Spaces of analytic functions on inductive/projective limits of Hilbert spaces Citation for published version (APA): Martens, F. J. L. (1988). Spaces of analytic functions on inductive/projective limits of Hilbert spaces. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR292007 DOI: 10.6100/IR292007 Document status and date: Published: 01/01/1988 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected]providing details and we will investigate your claim. Download date: 15. Nov. 2020
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Spaces of analytic functions on inductive/projective limits ofHilbert spacesCitation for published version (APA):Martens, F. J. L. (1988). Spaces of analytic functions on inductive/projective limits of Hilbert spaces. TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR292007
DOI:10.6100/IR292007
Document status and date:Published: 01/01/1988
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. IR. M. TELS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN
OP DINSDAG 8 NOVEMBER 1988 TE 14.00 UUR
DOOR
FRANCISCUS JOHANNES LUDOVICUS MARTENS
GEBOREN TE LITH
druk: wibro dîssertatiedrukker1j. helmond
Dit proefschrift is goedgekeurd door de promotoren:
Prof. dr.ir. J. de Graaf en Prof. dr. S.Th.M. Ackermans
Copromotor: Dr.ir. S.J.L. van Eijndhoven
Aan mijn ouders
CONTENTS
General introduction
I Preliminaries
§ 1. Locally convex spaces
§ 2. Inductive and projective limits
§ 3. Sequences and sequence sets
§ 4. Analytic functions
II Functional Hilbert spaces
Introduction
i
6
9
14
27
27
§ 1. Reproducing kemel theory 29
§ 2. Synnnetric Fock spaces as functional Hilbert spaces 44
§ 3. Examples of functional Hilbert spaces 55
Appendix 71
III Inductive and projective limits of semi-inner product spaces 77
Introduction
§ 1. Positive sequence sets
§ 2. Hilbertian dual systeins of inductive limits and projective limits
§ 3. Cross synnnetric moulding sets
Appendix
IV Spaces of analytic functions on sequence spaces
Introduction
§ 1. Generating sets
77
79
85
100
103
109
109
112
§ 2. Sequence spaces 120
§ 3. A sequence space representation of A(qi(]))) and A(w(]))) 127
§ 4. Elementary spaces in A(qi(]))) 134
§ 5. Compound spaces in A(q>(])))
Appendix
141
153
References 161
Index 164
Index of symbols 166
Summary 168
Samenvatting 169
Curriculum vitae 170
GENERAL INTRODUCT.ION
For a complex Hilbert space X, the so-called symmetrie Fock space
F (X) is defined as the direct sum of n-fold symmetrie Rilbert tensor sym
products of X. It is a topological completion of the symmetrie tensor
algebra of X. This Fock space, also called exponential Hilbert space, is
frequently used in quantum field theory by theoretical physicists and
can be represented as a functionalHilbert space S(X) with reproducing
kemel K(x,y) = exp(x,y)X, x,y € X. The Hilbert space S(X) consists of
analytic functions F on X with growth behaviour
2 IF(x) 1 ~ llFll8 (X) exp(illxllx) , x E X •
In a distributional approach of quantum field theory a symmetrie Fock
space construction has to be developed for locally convex spaces V,, e.g. test function spaces or distribution spaces, more general than
Hilbert spaces. In this general setting, it is not clear in which way
the symmetrie tensor algebra of V can be topologized or completed, a
difficulty that arises already at the level of the n-fold symmetrie
algebraic tensor products.
In the present monograph, these topological problems are solved rather
elegantly, constructing a 'symmetrie Fock space' of V which consists of
analytic functions on the (strong) dual V' of V. In this context we
observe that also the functional Hilbert space S(X 1) represents the
symmetrie Fock space F (X). sym
This thesis deals with a class of locally convex spaces, which admit
such symmetrie Fock space construction leading to locally convex spaces
of the same class. Our class consists of spaces which are inductive
limits of Hilbert spaces, projective limits of semi-inner product spaces
or both at once. The construction is carried out only for sequence
spaces and as a result we obtain locally convex spaces of analytic func
tions on sequence spaces. However, the concepts can be easily'.adapted
for more general locally convex spaces.
ii
The undèrlying treatise is based on an amalgamation of ideas from
- Aronszajn's reproducing kernel theory,
Bangmann's construction of Hilbert spaces of analytic functions,
which are repiesentations of the symmetrie Fock spaces F (ll:q) and sym F sym (R,2). general theory of analytic functions on locally convex spaces as
described by Nachbin and Dineen,
discussions on inductive/projective limits of (semi-) inner product
spaces by Van Eijndhoven, pe Graaf and Kruszynski. 1
Let us sketch our approach with an example. The spaces we start from in
this example are the so-called analyticity spaces and trajectory spaces
introduced in [G]. Spaces of these types be long to our class and have
been a major source of inspiration.
For a positive self-adjoint operator A in X, the continuous chain of
Hilbert spaces Xt, t E JR, is defined as follows. For t ~ O, Xt is the
space e-tA(X) endowed with the inner product (x,y)t = (etAx, etAy)X.
For t < O, Xt is the completion of X with respect to the inner product
(x,y)t = (éAx, etAy)X, There is a natural duality between the spaces
Xt and X_t. The set {Xt 1 t > O} is an inductive system with the ana
lyticity space SX,A as its inductive limit. The set {Xt 1 t < O} is a
projective system witb the trajectory space TX,A as its projective
limit. The spaces SX,A and TX,A are in strong duality.
In [EG 2] the spaces SFs (X),H and TFs (X),H are introduced for Ha suitable self-adjoint opr:ator associat~ with A. These spaces are
topological completions of the symmetrie tensor algebra related to SX,A and TX A' respectively .
• Besides, the analyticity space SFs (X),H and the trajectory space
TFsym(X),H admit a representation: spaces of analytic functions.
:_ndeed, in [Ma 1 ] the s paces SS (X), H and T S (X) , H are in troduced, where H denotes a positive self-adjoint operator in S(X) related to A. It bas
been proved tbat the space SS(X),H consists of analytic functions on
the strong dual TX,A of SX,A and tbat the space Ts(X),H consists of
analytic functions on the strong dual SX,A of TX,A'
iii
In this. thesis we replace SX,A and TX,A by more general inductive limits
V and projective limits W, which are in duality, and we construct spaces
of analytic functions on Wand V, respectively, similar to SS(X),H and
TS(X) ,H·
In the remaining part of this introduction we present a summary of each
chapter.
Chapter I contains prerequisites of the present monograph. We recall
results on locally convex spaces, in particular, results on inductive
and projective limits and results on analytic functions on locally
convex spaces. As an illustration serves the space of analytic functions
on the space of finite sequences. This function space is an important
tool in the aforementioned constructions.
Chapter II is devoted to Aronszajn's theory on reproducing kemels and
functional Hilbert spaces, We summarize results of Aronszajn's :theory
and represent the symmetrie Fock space F (X) as a functional Hilbert sym
space S(X). In a separate section we pay attention to examples of
functional Hilbert spaces, which are interesting for their own sake.
These examples are not relevant for the following chapters.
Chapter III is an exposition on inductive limits of Hilbert spaces and
projective limits of semi-inner product spaces. It contains a reformula
tion and a simplification of the theory as developed by Van Eijndhoven,
De Graaf and Kruszyóski. Starting from a Köthe sequence set and a
countable collection of Hilbert spaces, we construct an inductive limit
of Hilbert spaces and a projective limit of semi-inner product spaces
and describe their duality. Topological prope.rties of these inductive/
projective limits are put in correspondence with properties of the Köthe
set and the collection of Hilbert spaces .• In particular, certain Köthe
sets result in locally convex spaces, which are both inductive limits
of Hilbert spaces and projective limits of Hilbert spaces.
Chapters II and III contain the main tools for the final chapter, Chap
ter IV, and can be read independently.
In Chapter IV, for Köthe sets g, we introduce our sequence spaces
G. d[g] and G .[a], which fit in the description of Chapter III. The i.n proJ "'
by nonnegative sequences ä on countable sets ]). The projective limits + +
G • [g] arise from semi-inner product spaces R.2 [a;ID]. Here R.2 [a;ID] prOJ denotes the Köthe dual of R.2 [a;ID].
Applying reproducing kemel theory, for nonnegative sequences a on 1D
we introduce our elementary function spaces F. d[a] and F .[a]. The in proJ functions in Find [a] are analytic on R.;[a;1D] whereas the functions in
F .[a] are analytic on R.2 [a;ID]. Fora Köthe set g the collection prOJ {F. d[a] 1 a E g} turns out to be an inductive system and the collection in {F .[a] 1 a € g} a projective system. Our compound spaces F. d[g] and prOJ in F .[g] are defined to be the corresponding inductive and projective prOJ limits, The space F. d[g] consists of analytic functions on G .[g] and i.n · proJ yields a description of the symmetrie Fock space of G. d[g], Likewise, in the space F .[g] is a space of analytic functions on G. d[g] and . proJ in yields a description of the symmetrie Fock space of G .[g]. proJ
We charac terize the func tions in F • . d [g] and F . [g] wi th a growth in proJ
estimate and we characterize them also in terms of the coefficients in
their monomial expansions. The latter characterization leads to sequence
spaces G. d[up[g]] and G .[up[g)], homeomorphic to F. d[g] and in proJ i.n F .[g], respectively, with up[g] a Köthe set on multi-indices. prOJ Studying these sequence spaces, we obtain topological properties of the
corresponding compound spaces.
As an illustration of the theory we represent suitable infinite dimen
sional Heisenberg groups as groups of linear hcmeomorphisms on the
spaces F. d[g] and F • [g]. rn proJ
CHAPTER 1
PRELIMINARIES
In this chapter we present the prerequisites of the entire monograph.
§ 1. Locally convex spaces
In this section we present definitions and results on locally convex
spaces. Sources are the well-known textbooks [Yo], [Con] and [Sch].
(1.1) A aomplex veator spaae is a set V with two algebraic operations,
addition and scalar multiplication, which satisfy:
(V,+) is an abelian group.
- The scalar multiplication is a mapping from C x V into V for which
for all À,µ € C and v,w € V
À(v+w) • Àv + Àw
(À+ µ)v Àv + µv
(Àµ)v À(µv)
1v = v •
In general, we omit the adjective 'complex' in the expression 'a com
plex vector space'.
A t(lpological veator spaae is a vector space V endowed with a topology,
such that the mappings
(v,w)i+v+w, (v,w) E V x V
and
(À,v) 1+ Àv , (À,v) E C x V
2 I Preliminaries
are continuous with respect to the product topology. Hence for an open
set W in a topological vector space V the sets ÀW and v + W are open for
all À E t and v E V.
(1.2) A set W in a vector space V is
aonve;i; if \J Vv,wEW vÀ,0<À<1 Àv + (1-/,)w E W
balan,..ed if .., u " vvEW vÀ€(:, 1À1 ;l;l ÀV € W
absorbing if VvE V 3À>O : Àv E W •
(1.3) A locally aonvex apaae is a Hausdorff topological vector space,
such that the topology is locaUy aonve~, i.e. each neighbourhood of o
contains a convex, balanced and open set. We abbreviate 'locally convex
space' and 'locally convex topology' by 'L.C. space' and 'L.C. topology'.
A semino:rrn p on a vector space V is a function p: V + IR+, such that for
all À € t and v,w € V
p(v + w) :i p(v) + p(w) p( v)
1.1. Lemma
Let V be a (topological) vector space. Let W be a convex, balanced and
absorbing (open) subset of V. The Minkowsky functional or gauge kw• defined by
1 -1 Itw(v) = inf {À > 0 À v € W} , v € v
is a (continuous) seminorm on V.
1.2. Lemma ..
Let V be a (topological) vector space. Let p be a (continuous) seminorm
on V.
The set Up= {v € V j p(v) < 1} is a convex, balanced and absorbing
(open) subset of V.
The previous lemmas are the key to the main result in this section,
namely the relation between L.C. topologies and collections of seminorms.
I Preliminaries
(1.4) A collection of seminorms Pr is
{py j y E r} on a vector space V
sepaPating if for each v E V, v f. o, there exists y E r such
that py(v) f. 0 ,
di~eated if for each y1,y2 € r there exists y Er such that for
all v E V: p (v) ::î p (v) and p (v) ::î py(v). Y1 y Yz
(1.5) All subsets Win V, satisfying the condition:
For each w E W there exist .a finite set E c: r and s > 0 such that
w + n su c: w, yEE Py
establish a topology, T(V,pr)·
Let T denote a topology in a vector space V. The collection of semi
norms Pr on V gene~ates T if T = T(V,pr>·
Here follows the main result in this section.
1 . 3. Theorem
Let V denote a vector space endowed with a topology T.
3
Then V is a L.C. space iff T is generated by a separating collection of
seminorms.
Let Pr and qfi denote two collections of seminorms on a vector space V. The collections Pr and qfi are equivalent if Pr and qó generate the same
topology; to state it differently, the seminorms Py~ y E r, are contin
uous with respect to the topology T(V,qfi) and the seminorms q6, o E ó,
are continuous with respect to the topology T(V,pr).
The following result is useful.
1. 4. Lemma
Let Pr and qfi denote two directed collections of seminorms on a vector
space V. Then Pr and qö are equivalent iff
4 I Preliminaries
and
(1.6) A subset W of a L.C. space V is bounded if for each neighbour
hood U of o there exists À > 0 such that À-1w c U.
1. s. Lemma
Let W denote a subset of a L.C. space V. Then the set W is bounded if f for all continuous seminonns p on V sup {p(v) 1 v E W} < 00 •
In a L.C. space all compact subsets are closed and bounded.
The space of all continuous linear operators from a L.C. space V into a
L.C. space W will be denoted by B(V,W). We write B(V) instead of B(V,V).
1.6. Lemma
Let V and W denote L.C. spaces with generating directed collections of
seminorms Pr on V and q/:J. on W, respectively. Let A denote a linear
operator from V into W. Then A E B(V,W) iff
1. 7. Corollary
Let V and W denote L.C. spaces. Let B be a bounded subset of V and let
A E B(V ,W). Then A(B) is a bounded subset in W.
(1.7) The du.at of a L.C. space Vis the vector space of all continuous
linear functionals on V and is denoted by V'.
For each bounded subset W of V we define the seminorm -OW on V' by
-Ow(R.) = sup { 1R.(w)1 l w E W} , R. E V' •
I Preliminaries 5
The ûJeak dual of a L.C. space V is the space Vf endowed with the topol
ogy T(V', {-OW 1 Wis a finite subset of V}).
The weak dual of V is denoted by V&·
The strong dual of a L.C. space V is the space V' endowed with the
topology T(V', {-OW 1 W is a bounded subset of V}).
By VS we denote the strong dual of V.
Weak duals and strong duals of L.C. spaces are L.C. spaces.
(1. 8} For each v E V we define the linear functional .ev on V' by v
ev (i) = i(v) , v i E V'
The L.C. space V is reflexive if the mapping v i+ evv is a homeomorphism
from V onto (V$)e•
(1.9) A Cauchy net (vi)iEJ in a L.C. space V ie. 'a mapping j i+ vj from
a directed set J into the space V such that for each neighbourhood W of
o in V:
3.EJ V.EJ '<" v. - v. E W • l J 'l.~J l J
A Cauchy net (vi)iEJ in V has a limit v E V if for each neighbourhood W of o:
v-v.EW. J
The L.C. space V is aonrplete if each Cauchy net has a limit.
(1.10) The L.C. space Vis semi-MonteZ if each closed and bounded subset
. of V is compact. The semi-Montel space v· is MonteZ if V is reflexive.
( 1. 11) A c losed, convex, absorbing and balanced subset W of a L.C. space
V is a barrel. The L.C. space V is barreZed if each barrel in V is a
neighbourhood of o in V.
(1.12) The L.C. space V is bornologicaZ if every convex balanced subsèt
of V, which absorbs every bounded set in V, is a neighbourhood of o.
6 I Preliminaries
(1.13) A linear operator S from a Banach space V into a Banach space W is called nuaZear if there exist bounded sequences (v~)nEIN in VS and
(wn)nEIN in W and a sequence C € t 1 (IN) with nonnegative entries such
that
S(u) = l c(n)v' (u)w , nElN n n
u € v .
If V and W are Hilbert spaces, then the operator S is nuclear iff S is
a trace class operator.
Let V denote a vector space, p a seminorm on V and Np" { u € V 1 p( u) " 0}.
The normed linear space V is the quotient space V/Np with norm p de-"' p À
fined by p(v + N ) " p(v), v E V. By V we denote a completion of Vp and p À p we consider Vp as a subspace of Vp. Let q denote another seminorm on V
with VvEV : p(v) :îi q(v),
The.n VvEV: v + N cv+ N and p(v+N):;; q(v+N ). q ~ - p q
The canonical mapping jq,p: Vq +Vp is the uniquely determined continu-
ous mapping, which satisfies jq,p(v+ Nq) = v + Np, v € V.
A L.C. space V is nuclear if for each continuous seminorm p there exists
a continuous seminorm q such that
VvEV : p(v) :;; q(v)
and A A
the canonical mapping j V + V is nuclear. p,q q p
§ 2. Inductive and projective limits
In Chapter III we study dual pairs of L.C. spaces consisting of an
inductive li111it and of_ a projective limit. These inductive limits and
projective limits originate from directed collections of (semi-) inner
product spaces.
In this section we mention some generalities on inductive/projective
limits. For further references see [Sch], Chapter II, Sections 5 and 6
and [Con], Chapter IV, Section 5.
(2.1) Let V and W denote two topological vector spaces with V c W. The canonicaZ injection j from V into Wis defined by j(v) = v, v E V.
I Prelim.inaries
If the canonical injection is continuous, we express this by writing
V c;.. W.
2. 1. Deflnition
Let A denote a directed index set.
A collection of vector spaces· {Va ! o: E A}, each endowed with a L.C.
topology, is an inductive system if for each a,6 € A, with o: $ S,
Vac;..Ve.
2. 2. Deflnition
Let {Va 1 a € A} denote an inductive system,
7
The inductive limit, lim ind V , is the vector space U V,., endowed with o: E A 0: aEA "
the finest L.C. topology, such that Va c U V for each 13 E A. This "" aEA a
topology is called the inductive limit topology.
A neighbourhood basis B of o in lim ind V is formed by all convex o o: E A o:
balanced set U c U V~ such that for all ex E A the set U n V is open o:EA" · a in va. A criterium for continuity of mappings on inductive limits reads:
2. 3. Theorem
Let {Va 1 a E A} denote an inductive system. Let F denote a linear map
ping (seminorm) from lim ind V into a topological vector space (IJ (into o: E A
.IR+).
Then F is continuous iff for each S E A the linear mapping (seminorm)
FIVa is continuous on v8•
Proof:
Cf. [Con], Section IV.5.
The second part of this section deals with projective limits.
2.11. Deflnitioh
Let A denote a directed index set.
A collection of vector spaces {Va 1 o: E A}, each endowed with a L.C.
topology, is a p~ojective system if for each o:,S E A, with a $ $,
Vac;..Va.
c
8
2.5. Definition
Let {Va 1 a E A} denote a projective system.
The projeative Zimit, lim proj V , is the vector space
I Preliminaries
n v endowed a€A a a E A et
with the coarsest topology, such that n v c;_ va etEA et
for each 8 E A. This
topology is called the projeative limit topoZogy.
We remark that for the
of v E n V is given etEA a
projective limit topology a neigbbourhood basis
by all intersections fl F 1 (U ) , where U,.... is a aEE a a "'
neighbourhood of v in va and where E is a f inite subset of A.
The next results are not very surprising.
2.6. Lemma
Let {Vet 1 a E A} denote a projective system where each Va is endowed
with topology T(Va•Pra). Let r • a~A ra and let Pr = {py restricted to fl V j y E r}. Then
a.EA a
a. The topology of lim proj V,,,,, is equal to T( n V,...., Pr). aEA "' aEA....,
b. The space lim proj Va is a L.C. space iff the collection Pr is , et E A ·
separating.
Proof:
See [Sch], Chapter II, Section 5.
2. 7. Theorem
Bet {Va. 1 a E A} denote a projective system. Let F denote a linear map
ping from a topological vector space W into lim proj V • a. E A a
Then F is continuous iff f or each a. E A the mapping F is continuous
from W into V ó'.•
Proof:
See [Sch], Chapter II, Section 5.
As is well known, eacb complete L.C. space is a projective limit of
Banach spaces.
In the next section we give two classica! examples, one of an inductive
limit and one of a projective limit. See Lemma.3.t.
From Lemma 2.6 it follows that not every projective limit is Hausdorff.
In general, it is hard to check whether an inductive limit is Hausdorff.
IJ
IJ
i' ;1
I Preliminaries
§ 3. Sequences and sequence sets
Throughout this section, Il denotes a fixed, countable set. Functions
from Il into t will be called sequences (labeled by Il) and tbey are
denoted by a,b, "., etc. By w(ll) we denote the set of all sequences
labeled by ll.
For each subset JE of U the sequence Xm is defined by
j e: JE
j t E
So Xm is the characteristic function of the subset E. In particular, we errploy the notation n = Xn• 0 = x0.
Fix u € w(Il). The suppor-t of u, which we denote by ll[U], is the set
{j € Il i u(j) r O}. lts complement is denoted by llc[u], viz.
Xc(U] = {j E U ! U(j) = O} •
Correspondingly we have the functions nu and Ot1•' defined by
and
Hence for all j e: l[
We mention the following operations on sequences:
Let u, v E w(lI), À e: t:.
Add.ition. The sequence u+ v € w(X) is defined by
[u + v](j) = u(j) + v(j) , j e: Il •
Saa'lar multipliaation. The sequence ÀU e: w(X) is defined by
[ÀU](j) = À[U(j)) , je:x.
Pl'odwt. The sequence u • v € w(lI) is defined by
[u • v](j) = u(j)v(j)., j e: u •
9
10 I Preliminaries
Thus w(ll) becomes a commutative algebra.
Tensor product. Let w E w(Il) with JI a countable set. The sequence
n1'· (x,y)2n • [nf2] r(n-2m+lq+1) M (x,y)lxl2mlyl2m •
m=O 22m r(n-m+iq+1) q,n-2m 2 2
Proof:
Use relation (3.12).
c
IJ
IJ
68 II Functional Hilbert spaces
Let the m-th term in Corollary 3.23 be denoted by N (x,y). As sets 2 2 q,n,m
we have H(lRq,N ) = {x,... lxl2m G(x), x E lR 1 G E H 2m}. With the q,n,m q,n-
aid of property (3.8) it can be shown that
[n/2] H(lR2 ,L ) = 19 H(lR\N )
q,n m=O q,n,m
This relation expresses the well-known fact that each n-homogeneous
polynomial Pn can be written as
Pn(x) 2 2[!n]
~(x) + lxl Qn_2(x) + ••• + lxl ~-2 [!n](x)
where Q 2 ., 0 ~ j ~ [!n], denotes a harmonie, (n-2j)-homogeneous polyn- J
nomial. Cf. [Vi], Chapter IX, Section 4.
Let O(q) denote the orthogonal group on lRq and let IT be the repre-q,n -1 .
sentation of O(q) in H(lRq,L ) given by II (R)F = F o R , R € O(q), q,n q,n F € H(lRq,L ). It is clear that the spaces H(lRq,N ) remain in-q,n q,n,m variant under the unitary operators R»- Il (R), R € O(q). q,n
The representations Rt+ IT (R)IH(lRq N )' R € O(q), are irreducible. q,n ' q n m
Cf. [Vi], Chapter IX. ' '
As a corollary to Theorem 3.22 we mention
3. 24. Corollary
Let q 3 and n ;;; O. Let F E H For all x E lRq q,n
IF<x>l2 llFll~ rog-1 )f(n+g-2)
lxl 2n ~ r(n+iq-1)r(q-2) 2n ' q n.
Proof:
For all x E lRq we have IF(x)l 2 ~ llFll~ M (X,x). The result.follows q q,n
from Theorem 3.22 and relation (3.10) with À= iq-1. a
The next theorem presents an explicit expression for the reproducing
kemel M of the functional Hilbert space H . q q
II Functional Hilbert spaces
3. 25. Theorem
Let q ~ 3. For all x,y E IRq we have
x"' 0 'Î' y
Mq(x,y) = 1 , x = o or y = o •
Proof:
. "" ~"" Since H = ti 0
H , Lemma 1.11.c. implies that M = M q n= q,n q =O q,n •
Let F E Hq. Since F E H(IRq ,Lq), we have j F(X} j i llFllH exp(!jx I~), x € IRq. We give another uniform estimate on m.q. q
3. 26. Corollary
Let q ~ 3.
For all p > 1 there exists y > 0 such that q,p
Proof:
Let F E Hq and let X E IRq. Bij relation (3.1.0) we get that
00 lx1 2n l jF(x} 1 s llFll (1 r(!q-1}r(n+q-2) _2_) •
- H =O r(n+lq-1)r(q-2) 2n 1 q n.
r(!q-1)r(n+q-2) < 2 pn Let p > 1. There exists Yq,p > 0 such that r(n+tq-l)r(q-2) Yq,p •
69
c
Hence the assertion follows. c
Finally, we consider harmonie functions in 800
, the Bargmann space of
infinite order. We shall prove that U H is dense in 800
• We recall
that the operators T and Z have bee~lnt;oduced in Definition 3.7. q q
3. 27. Definltion
A function F € 800
is called a harmonia funation if there exists q E JN
such that F = Zq(F) and the restriction FiIRq is harmonie on IRq.
70 II Functional Hilbert spaces
3. 28. Theorem
The space of all harmonie functions in 800
is dense in 800
•
Proof:
For q ~ 2, let
If U V is dense in H(lRJl.2 ,L ). then Lemma 3.12 implies the statement. q2:2 q co
With reproducing kernel theory we prove that indeed U V is dense in q~2 q
H(lRt2,L,).
Let M be the reproducing kemel of v . Since vq c vq+1 c H(lRt2•L,x,), q ,..., ~ q
we find that Mq ::> Mq+l ::> L00 • Because F 1+ Fimq• FE Vq' is a unitary
operator from V onto H , we obtain that q q
00
Mq(x,y) = M (T x,T y) l M (T x,T y) q q q n=O q,n q q
Fix x,y E lRt2 and n ;s; 0. Corollary 3.23 impliès that
M (T x, T Y) :;; -1, IX 1 n2
• IY I n2 q,n q q n.
and Theorem 3.22.b. implies that
1 n lim M (T x, T y) = n! (X,y) 2 q-><x> q,n q q
Hence
By Theorem 1.22 the union U V is dense in H(lRt2,L00).
q§!2 q
In [Ma 11 the following results can be found. They are based on the
previous theorem.
(3.13) A generalization of the Mean-Value Theorem for harmonie func
tions: For all F € H(lRJl.2,L"',), X E 1Rl2
and r > 0
F(X) = lim - 1- /· F(T X + r~) do 1 (~) •
q-><x> oq-1 q q-S q-1
(Cf. [Ma 1], Theorem 3.22.)
tJ
II Functional Hilbert spaces
(3.14) A Weak version of the min-max principle of harmonie functions:
Let D be a bounded open subset in lRR-2 with boundary r such that D is
weakly closed and let F E H(lRt2 ,L~). Then
VXED IF(x)I ~ sup IF(y)I • YEf
Cf. [Ma 1], Theorem 3.15.
(3.15) A construction of an orthonormal basis of harmonie polynomials
in B~. Cf. [Ma 1], Definition 3. 17.
Appendix
In this appendix we consider some linear operators in the symmetrie
Fock space S(H) of a Hilbert space H. The operators we study, are the
creation and annihilation operators, dilatation operators and second
quantized operators. Reproducing kernel theory plays an important role
in this appendix. We feel inspired by [Ba 1].
Recall that Sn(H) = H(H, ~! (•,•)~), n ~ O, and S(H) = H(H,K), where
71
K = exp[(•,o)H]. As usual, V(H,K) denotes the linear span <{Ky 1 y € H}>.
We start with the annihilation and creation operators•
A. 1. Deflnltion
Let u € H.
The annihilation operatoP a(u) in S(H) is defined by
[a(u)F](x) = lim (F(x+Àu) - F(x))/À, À-t-0
where F is in the maximal domain D(a(u)).
The aPeation opeI'atOP y(u) in S(H) is defined by
[y(u)F](x) = (x,u)H F(x) , x E H ,
where F is in the maximal domain D (y(u)).
x € H ,
72 II Functional Hilbert spaces
REMARKS:
(A.1) For all u E H the space V(H,K) is contained in D(a(u)) as well
as in D(y(u)),
(A.2) Let u E H. The annihilation operator a(u) maps Sn(H) into
S 1
(H) (o:(n) annihilates a particle) and the creation operator y(u) n-map s S (H) into S
1(H) (y(u) creates a particle). With reproducing
n n+ kemel theory it can be shown that
\>'FES (H) n
(A.3) For all u,v E H we have the Canonical Commutation Relation
a(u)y(v) - y(v)a(u) = (u,v)H I ,
where I denotes the identity on S(H).
(A.4) For all u,x € H we have
(a(u)F) (x)
(y(u)F) (x)
A.2. Theorem
Let u € H.
(F • y(u)Kx) S (H)
(F ,o:(u)Kx) S (H)
F E D(a(u))
F € D(y(u))
The operators o:(u) and y(u) are mutually adjoint.
Proof:
We only give a sketch of the proof.
By (A.4) we derive D(y*(u)) cD(a(u)) and D(a*(u)) c D(y(u)).
Finally we introduce the notions of Köthe set and moulding set. Bach
Köthe set yields an inductive limit and a projective limit of weighted
i 2-spaces. A moulding set is a Köthe set of a special type. The defini
tion is taken such that the corresponding inductive limits and projec
tive limits have manageable topological properties. But first we intro
duce multipliers.
a
84 !II Inductive and projective limits
1. 11. Definition Let pc: w+(l[).
A sequence C: is called a p-multiplieP if C: • p ,.., p.
1.12. Lemma
Let ë; be a p-multiplier for some pc: w+(Il). Then
a. The sequence C:-l is a p-multiplier.
b. If :O:[l;] =,Il, then ë'; and C:- 1 are alsi> p 11-multipliers.
Proof:
a. For each a € p we have 1I [a] c: 1I [ë';] • -1 Let a € p. There exists b E p with <: • a ;s b, whence a :S l; b.
Conversely, let c E p. Then there exists d € p with c ;s l; • d and
therefote <;-1 c :., d. So p ..... <;-1 • p.
b. let ll[l,;] =ll. Then (ë'; • p)# = <;-1 • p11• Since l; • p,..., p, we find that ?-l # # Th 7 -l . # 1 . i· S . .,, • p ,..., p • e sequence .., is a p -mu tip ler. tatement a. im-
plies that <: is a p#-multiplier. o
1. 13. Definition
A sequence set p in w+(ll) is a Köthe set if pis separating and quasi
directed. + A Köthe set pis a moulding set if p admits a p-multiplier <:in t1
(:n:).
1.111. Lemma
Let p denote a moulding set.
Then p# is a moulding set,
Proof:
Let <:: denote a p-multipl_ier with <; E t~ (Il.). Since p is separating, we
get Il[{) =Il. From Lemma 1.12 it follows that <:is a p#-multiplier. o
Our concept of Köthe sets is a modification of the one in [KG].
III Inductive and projective limits 85
§ 2. Hilbertian dual systems of inductive limits and projective limits
In this section we introduce dual systems of inductive limits and pro
jective limits. Here we feel inspired by Van Eijndhoven and De Graaf,
[EG 1 l, [EG 2] and [E 2]. The presentation here differs fran the presenta
tion in the fore-mentioned references in order to make the topological
structure of our inductive limits and projective limits more explicit.
Throughout the remaining part of this chapter a f ixed sequence of non
trivial Hilbert spaces (Hm)mE:n: is considered. Here n: denotes a count
able set again.
2. 1 • Defini tion
Let XHk denote the Cartesian product k~ ~ endowed with the product
topology.
Elements h of XHk are mappings on :n: with h(m) E Hm' m E Il, and are
occasionally denoted by (h(m)) E.,,.. Fix m E 1[. We identify h E H m,,.. m m with (h o ) En: € XHk. So the Hilbert space H is considered as a ro m,n n m subspace of XHk.
The projection P : XHk + H is defined by P h = (h(m)ó . ) E:D:' h E XHk. m m m m,n n
Let H denote the Hilbert space {h E XHk 1 ~Il llPm hll~ < oo} with inner
product (h,g)H = Î-mEn: (Pm h,Pm g)lfui.
The operators Pml H' m E 1I are mutually orthogonal projections on H and
we have Î-m€1[ Pm h = h, h E H. So H = 4Ï\€1[ H k' Note that the operators
Pm, m E Il, are continuous on XHk.
2.2. Definition
Let a € w + (E).
The linear operator Aa: XHk + XHk is defined by Aa = ~lI a (m)Pm.
Let a,b € t/(u) with a :;;; b. The operators AalH and Abltt are positive
self-adjoint operators in H and i.satisfy the operator inequality
86 III Inductive and projective limits
Two types of seminonns will be used. These seminorms are defined on
subsets of XHk.
2. 3. Definition
Let a E </ (lI).
The seminonns Pa and qa are def ined by
Dom(pa) = {h E XHk 1 Aa h E H} ,
\ ! Pa (h) = 111\a hl!H = (~JI lla(m) Pm hll~) , h E Dom(pa) ,
and
q (h) = I lla.(m) P hllH , h E Dom(qa) • a mElI m
2.4. Lemma
Proof:
Let h E Dom(Pi,)· Using Hölder's inequality we get
Î Ha(m)P hllH= I (a·b-1)(m)llb(m)P hllH~
mElI m mEl! m
We recall tbat T(V,pr) denotes the topology on a space V, which is
generated by a collection Pr of seminorms on V.
0
III Inductive and projective limits 87
2. s. Corollary
Let V denote a subspace of XHk' Let p c w +(ID) be a directed set such -1 +
that V c n Dom(p) and v E 3bE : :n:[a] c Il[b] and a • b E Jl2(Il). aEp a a p p
Then V c n Dom(q ) and the topologies T(V,q ) and T(V,p ) are the aEp a P P
same.
Proof:
From Lemma 2.4 it fellows V c n Dom(4 ). So for each a E p there aEp a
exists b € p such that
Since p is directed, the collections of seminorms p and q are directed p p
too. Because of Theorem I.1.4 the above mentioned topologies are the
same.
After these preliminary results we are ready to introduce inductive
limits. Building blocks for these limits are Hilbert spaces.
2.6. Definition
LetaEw+(D:).
The Hilbert space H[a] is defined to be the space Aa H endowed with the
inner product
(h,g)H[a] (A -1 h,A _lg)H • a a
h,g E Aa H •
2. 7. Lemma + Let a,b E w (Il), The following results are equivalent:
a. a ~ b •
c. H[a] c;..H[b] •
Proof:
c. "b. This implication is trivial.
-1 b. "a. Statement b. implies Ab_1 H c H. Hence b • a E R.
So I[p;Hk] c::: Dom(pb), b € p#. # I[p;Hk] c: Dom(qb), b € p •
The topological considerations are based on the next lemma.
2.11. Lemma # + Let p denote a KÖthe set such that p • p c: t
1 (lI). Let V denote a
balanced convex subset of I[p;Hk]' such that V n H[a] is a neighbour
hood of o in H[a] for each a € p.
Tben there exists U € p# such that
90 II! Inductive and projective limits
Proof:
Let kv denote the gauge of V. For all h E I[p;Hk] the inequality
kv(h) < 1 implies that h E V. Since V n H[a], a E p, is a neighbourhood
of o in H[a}, there exists µa > 0 such that
For all m € ll, a € p and h € Hm n H[a} we have
UhBH = a(m)llhllH[a] •
Since pis separating, the restriction kvlfiui is continuous at o for
each m € ll. So, we can define the sequence u € w+ (ll.) by
u(m) = sup {kv(h) 1 h E Hm' llhllH = 1} •
We prove that u E p#. Let a E p. Then for all m E lI
a(m)u(m) ;;> sup {a(m)kv(h)/llhl!H 1 h € Hm' h ;. O} ;;>
;;> sup {kv(Aa h)/llAa hHH[a] 1 h E H, Aa h ;. O} ;;> µa •
Hence u E p#.
Let h € I[p;Hk] with qu (h) = ~Ell u(m) llPm hllH < 1. We will show that
h E V. Let h € H[a] for some a E p. Since the seminorm kviH[a] is
continuous at o, i t is continuous on H[a]. The series ~Ell Pm h con
verges to h in H[a]-sense, hence
kv.<h} ;;> i: kv(P h) ;;> I u(m)llPm hllH < t • mE:lI m m€][
So h E V.
2.12. Theorem
Let pc: w+(ll) be a Köthe set.
a. Then T2 = T( U H[a],p #) is contained in TI, the inductive limit aEP P
topology of I(p;Hk]' and so I[p;Hk] is a Hausdorff space.
b. lf in addition p • p# c: ~~ (JI}, then T I is contained in
T1 = T(a~p H[a],qp#) and so T2 c: Tz c: T1•
c
III Inductive and projective limits
Proof:
a, Let u E p#. For all a E p the seminorm Pu IH[a] is continuous. So
Theorem I.2.3 implies that A .is continuous with respect to the . u
inductive limit topology T1.
91
b. Let p • p# c i7 (lI). Let V denote an open, convex and balanced neigh
bourhood of o in I[p; f\.;1· Hence V 11 H[a] is a neighbourhood of o in
H[a] for each a E p. Because of Lemma 2.11, there exists u E p#,
such that {h E I[p; \1 1 <Zu (h) < 1} c: V. Hence V is a neighbourhood
of o with respect to T1
•
Moulding sets p c w+ (E) have the property that (J • p# c R-7 (E).
2. 13. Corollary
Let pc: w+(E) be a moulding set. Let T2 , T1
and T1 be defined as in
Theorem 2.12.
Then we have T2 " T 1 "" T1•
Proof:
Since p • p# c Q,7 (n:) we have T2
c T I c T1• By assumption there exists
a p-multiplier C: E i7 (n:). For each a E p there exists b E p such that
C1 • a :S b, whence ll:[a] cE[b] and a • b-t :S (. Corollary 2.5 yields
that r 2 " T1, whence the three mentioned topologies are equal.
The previous corollary can be extended. A Köthe set p, which satisfies
3 v 3 : C 1 • a ::., b • <:n7 (lI) ,n.[<:1 =lI aEp bEp
is in fact a moulding set (with p-multiplier Ï:).
For a Köthe set p, which satisfies the weaker condition
# + we also have that p • p c i 1 (Il.) and that Corollary 2. 5 implies that
the three topologies, mentioned in Theorem 2.12, are equal.
We state three more results on the inductive limits I[p;Hk] where p is
a moulding set.
a
0
92 !II Inductive and projective limits
2.111. Theorem + Let p,crc: w (II) be moulding sets. The following statements are equiva-
lent:
a. 1 [p;Hk] c;. 1 [cr;Hk] •
b. I[p;Hk] c: I[cr;Hk) as sets.
c. p ~ a •
Proof:
a. • b. This implication is trivial.
b. • c. Let <: E R.1 (II) denote a p··multiplier. Let a E p and let h E XHk
with llPm hllH = 1, m E lI. Then Aa(h) = Az:-1.a<Az: h) E I[p;Hk). Hence
there exist b E cr and g E H. such that Aa(h) = Ab(g). So for all
m E II
a(m) = b(m)llPm gllH ,
whence a ::, b.
c. "° a. LE!l!lllla 2.9.a. yields this implication.
2.15. Corollary
Let p,cr cw+(II) be moulding sets.
Then I[p;Hkl = I[o;Hk] iff p,..., cr.
The next theorem can be found in a similar form in [Ma 2].
2. 16. Theorem
Let p c: w +(II) be a moulding set and let d (m) = dim(Hm), m E lI.
Then the space I[p;Hk] is nuclear iff
+ VmE][ : d(m) < oo and "aEp "'ue:p#: a • u • d E R.1 (II) •
We remark that the condition a • u • d E i~ (II) is equivalent with the
condition that A ltt is a trace class operator on H. a•u Proof:
Let w E p#. The Hilbert space H[w-1] is a completion of the quotient
space 1 [p;Hk]/p;(o).
0
III Inductive and projective limits 93
Now I[p;Hk] is nuclear iff for each u E p# there exists v € p# with the
properties:
(2.1) u ~ v '
(2.2) -1 -1 The canonical injection An : H[v ] c;..H[u ] is nuclear.
u -1
Since A _1 and Au are bounded operators from H into H[v ] and from -1 v
H[u ] into H, respectively, condition (2.2) can.be replaced by
(2.3) The operator Au·v-l is a trace class operator on t/.
Note that trace(J\ _1) = I.' (u • v-1) (m) trace(P ) u•v m~JI m
". Let a E p and u E p# By assumption there exists v E p# such that
u ~ v and Au·v-l is a trace class operator on tl. So
Since p and p# are separating, all d(m) are finite •
.-. Let l; E i7 (JI) denote a p-multiplier. Let U E p#. Put V = d • ë;-1 • u.
Then u;:;, V and the assumption implies that v E p#. Finally we have
I.' -1 trace(A -1) = t. ê;(m)d (m) trace(P ) u•v mEJI m
Next we consider projective limits. We construct these projective
limits from semi-inner product spaces.
2. 17. Definition
Let a E w+(JI).
The semi-inner product space H+[a] is defined to be the vector space
endowed with the semi-inner product
+ h,g E Aa (ff) •
The space tl+[a] is a Hilbert space iff JI[a] =II. In that case
tl+[a] = H[a -l].
0
94 II! Inductive and projective limits
2.18. Lemma + Let a,b E w (][).
Then a ~ b iff ff+[b] c;.. ff+[a].
Proof:
•· Let h € H+[b]. Then
So ff+[b} c:;.ff+[a].
". There exists y > 0 such that Vh€H""[b] : IAa hi 2 ::á YIAb hi2•
Since H c: H+[b], we find that a(m)';;î yb(m) for all m EU. Soa ;S b. c m
Fora Köthe set pc: w+(lr) the set {H+[a] 1 a € p} is a proji;ctive
system. Cf. Definition I.2.4.
2. 19. Definition
Let pc: w+(1l) be a Köthe set.
The projective limit P[p; (fik) kEII] is defined by
Lemma I.2.6 implies that P[p;Hk] is the space n H+[a] with the topo-
b h • . ae:p . . h
logy generated y t e seminorms Pa• a € p. Since p is separating, t e
space P[p;Hk] is Hausdorff.
2. 20. Theorem + Let p,o c: w (11) be Köthè sets. Then
Proof:
a. Let p < cr. Lemma 2.18 implies n H+[a] c: bn H+[b]. Since for each ~ a€o Ep
a E p there exist y > 0 and b E o such that
III Inductive and projective limits
we get that P[a;Hk] ~ P[p;Hk].
Conversely, let P[a;Hk] ~ P[p;Hk]. Continuity of the injection
implies that for each a E p there exist y > 0 and b E a such that
Pa(h) ~ ypb(h) for all h E P[a;Hk]. Since Hm c H, m E 11, we find
that a ;;;; yb.
b. We remark that p ~ a iff p :;;, a and a :;;, p.
Next we consider some topological properties of the projective limits
P[p;Hk]. For definition see Section I.1.
2. 21. Theorem + Let p c w (11) be a Köthe set.
The space P[p;~] is complete.
Proof:
The proof is standard. Let (hi)iEJ denote a Cauchy net in P[p;Hk].
For all m E 11, the net (P h.) .EJ is a Cauchy net in H with limit h • m l. l. m m
95
D
Let h E Xflk with P H = h • For each a E p the net (A b.) 'EJ is a Cauchy m m a i i
net and bas limit ha € H. It turns out that ha = Aa h.
Hence h € P[p;Hk] and h is the limit of the Cauchy net (hi)iEJ" a
2. 22. Theo rem
Let pc w+(ll) be a moulding set.
For all b E p we have the inclusion En H+[a] c Dom(qb) and the projec-. a P ,...
tive limit topology of P[p;Hk] is equal to T( n H [a],q ). aEp p
Proof:
The projective limit topology equals T( En H+[a], p ) • Since the set p a P P
is moulding, for each a E p there exists b E p such that a ~ b and
a • b-1 E R-2(11). Corollary 2.5 implies the statement. o
In the following theorem we characterize bounded sets in P[p;Hk].
2. 23. Theorem
Let p cw+(ll) be a moulding set. Then
96 Ill .lnductive and projective limits
a. For all u E p# and all bounded subsets 8 of H the set AU B is bounded
in P(p;Hk].
b. For a bounded subset W in P[p;Hk] there exist u E p# and a bounded
subset B of H such that Au maps B hoineomorphicly ànto W with
respect to the relative topologies.
Proof:
a. Let u E p# a.nd let 8 be a bounded set in H. For all a E p and h E B
we have
So J\u 8 is bounded.
b. Let W be a bounded set in P[p;Hk]. Let ~ E i7(lI) denote a p-multi
plier. For m E lI put
-1 1 u(m) • ~ (m) • sup {llPm hllH h E W} .
Since p is separating, U(m) < oo for all m € Il.
Let a E p. For all m E :U: we have
a •u(m) = sup {i'.;-1 (m)a(m)llPm bllH 1 h E W} ::1
::1 sup {q~-t·a(h) 1 h € W} < oo.
Hence u E p#. Further, for all h E W we have An h = h and for all . u
m E lI llPmAu-1 hllH ::1 ~(m). Put B = {Au-1 h 1 h E W}. Then Bis a
bounded set in H, the set W equals A0
B and J\u: B + W is a bijection.
Naw we prove that Au is a homeomorphism. Fix h E B. For a;n g E B
and a E p we have pa(./\u(h-g)) ::1 la• ul00
• Nh-gllH' So ./\u maps 8 con
tinuously onto W. Finally we prove continuity of ./\u-t from W onto 8. Let h E W. For
all g E W and finite subsets lF of :0:
III Inductive and projective limits 97
-1 + Since {u • XIF} ;;;, p and ?.; E t 1 (Il) , we get that 11.u-1 llîaps W con- .
tinuously onto 8.
2. 2~. Corollary
Let pc: w+(E) be a moulding set and let W denote a subset of P[p;Hk].
Then W is compact iff there exist u E p# and 8 c: H, 8 a compact subset,
such that W = 11.u B.
2. 2 5. Corollary
Let p c w +(Il) be a moulding set and let h E XHk.
Then h E P[p;flk] iff there exist u El: p# and g E H such that h = Au g.
So P[p;Hk] = I[l;Hk] as sets.
We recall that a L.C. space in which all bounded and closed subsets are
compact, is called a semi-Montel space.
2. 26. Theorem
Let pc: w+(Il) be a moulding set.
Then the space P[p;Hk] is semi-Montel iff VmEil dim(flm) < 00 , i.e. Pm
is of finite rank.
Proof:
=>. Let B denote the closed unit ball in H • Then B is a bounded and m m m
closed subset of P[p;Hk] and therefore a compact subset of P[p;Hk].
Hence Bm is compact in H and so dim(fl ) < 00• m m
~. Let W denote a bounded and closed subset in P[p;flk]. Let ?.; E i~ (Il),
u E p# and B c: H be the same as in the proof of Theorem 2.23.b. For
all h E 8 and m E :0:, llPm hll H :ii l.;(m). Since all Hm are fini te dimen
sional, the set B is compact. Because 11.u is a homeomorphism from B onto W, the set W is compact,
In the final part of this section we define a pairing between the in
ductive limit J[p;flk] and the projective limit P[p;Hk]. The pairing is
sesquilinear.
2. 27. Definition
Let pc w+(:U:) be a Köthe set.
By a Hilbertian dual system we mean the pair of spaces I[p;Hk] and
0
0
98 III Inductive and projective limits
P{p;ffk] with their pairing <•,•>, defined by
with a E p such that h E H[a].
The pairing in Definition 2.27 yields a representation for the duals of
our inductive/projective limits. First, however, we introduce classes
of linear functionals.
2.28. Definition + Let p c: w (lI) be a KÖthe set.
For g E P[P;f\l we define the linear functional ~g on I[p;Hk] by
~g(h) = <h,g>, h E I[p;ffk].
For h E I[p;HkJ we define the linear functional 'l'h on P[p;Hk] by
'l'h(g} = <h,g>, g E P[p;Hk].
For h E H we define the linear functional rh on H by fh(g)
g E ff.
(g, h) H'
2. 29. Theorem + Let p c::w (ll) be a Köthe set. Letland m denote linear functionals on
I[p;Hk] and P[p;Hk]' respectively.
a. The functional l is continuous iff there exists g E P[p;Hk] with
t = ~ • g
b. The functional m is continuous iff there exists h E I[p;Hk] with
m = 'Vb.
Proof:
a. •. Since t o PmlH is continuous, there exists 8ni E Hm such that
l 0 PmlH = r &!' m E ll. Let g E XHk with Pm g = ~· Fixa E p. The restriction lltt[a] and hence l o Aaltt• is continuous.
So there exists fa E H such that to AalH = rfa' We find fa= Aa g.
Bence g E P[p;Hk] and l = ~g· .
""'· Let l = ~g for some g E P[p;Hk].
For all a E p we have
III Inductive and projective limits 99
whence llH[a] is continuous. Sol is continuous.
b. "*• Since m is continuous, there exist µ > 0 and a E p such that
Hence m 0 Aa-1 IH is continuous and there exists f E H such that ----
m o Aa-11 H = r f" For g E P[p;Hk] we have m(g) = (Aa g,f)H = <Aa f,g>.
So m = '!' (A f). a
<=. Let m = '!'(Aa f) for some a E p and f E H. For g E P[p;Hk] we have
lm<g) 1 = 1 <A f ,g>I a
So m is continuous.
We finish this section with two theorems on strong duals. Only sketches
of their proofs will be given.
2. 30. Theorem
Let pc w+(Il) be a moulding set.
Then the mapping '!': I[p;Hk] + (P[p;Hk])~ is an antilinear homeomorphism.
Proof:
Theorem 2.29 yields that '!' is an antilinear isomorphism from I[p;Hk]
onto (P[p;Hk])'. The tii>pology of (P[p;Hk])S is equal to
where B is the unit hall in H and where
sup ll<Au h) 1 , hEB
# Now, for each U E p and h E I[p;Hk] we have
a
a
100 111 Inductive and projective limits
2. 31. Theorem
Let pc w+(I) be a moulding set.
Then the mapping <P: P[p##;Hk]-+ (l[p;Hk])S is an antilinear homeomorphism.
Proof:
The proof is similar to the previous one. We remark that
and the topology of (I[p;Hk])~ is equal to
where Be = B n \l>(ID). c
§ 3. Cross symmetrie moulding sets
In this section we investigate which topological conditions on the
spaces I[p;Hk] and P[p;Hk] are equivalent with the #-symmetry condition
on the moulding set p.
3. 1 . Theorem
Let pc w+(I) be a moulding set.
The following statements are equivalent:
a. The set p is #-symmetrie.
b. I[p;Hk] ##
I[p ;Hk]
c. P[p;Hk] ##
P[p ;Hk]
d. I[p;Hk] #
P[p ;Hk]
P[p;Hk] # e. I [p ;Hk]
Proof:
a. • b. See Corollary 2.15.
a. • c. See Lemma 2.20.b.
III Inductive and projective limits
a. * d. Observe that p# = p###. Corollaries 2.13 and 2.25 yield that
P[p#;Hk] = l[p##;Hk].
101
# ## a. * e. Corollaries 2.13 and 2.25 yield that l[p ;Hk]=P[p ;Hk]. a
For the notions in the second theorem we recall Section I.1.
3. 2. Theorem
Let p E w+(JI.) be a moulding set.
The following statements are equivalent:
a. p is #-symmetrie.
b. P[p;Hk] is reflexive.
c. I[p;Hkl is reflexive.
d. P[p;Hk] is barre led.
e. P[p;Hk] is bornological.
f. I[p;Hk] is complete.
Proof:
a. * b. From Theorems 2.30 and 2.31 it follows that the spaces ## ( ') 1 • ## P[p ;Hk] and (P[p;Hk])S S are homeomorphic. Hence P[p ;Hkl
= P[p;Hk] iff P[p;Hk] is reflexive.
a. * c. Since (P[p;HkDé = (P[p##;HkD~" Theorems 2.30 and 2.31 imply
that the spaces I[p##;Hk] and ((I[p;Hk])S)~ are homeomorphic. Hence
l[p##;Hk] = I[p;Hk] iff I[p;Hk] is reflexive.
a • .,. d.,e. By Theorem 3.1.e. and Theorem 2.10 these implications follow.
a • .,. f. By Theorem 3.1.d. and Theorem 2.21 this implication fellows.
d. "*' a. Let a E p##, Let W denote the set
{h E P[p;H ] 1 sup {a(m)llP hllH 1 m E JI.} ::l 1} m m
Then W is a barrel, because W is a closed, convex, balanced and
absorbing set. Hence W is a neighbourhood of o in P[p;Hk]. So there
102 III Inductive and projective· limits
exist µ > 0 and b € p such that
Bence a ~ b.
e ..... a. Let a € p##. Let W = {h € P[p;Hk] J qa(h) < 1}.
Thenlll is a convex and balanced subset of P[p;Hk]. It is easy to see
that W absorbs every bounded subset of P[p;Hk]. Hence W is a neigh
bourhood of o in P[p;HkJ. Further, see the proof of d. • a.
f. ... a. Let a E p#ll, let (Jiq) qElN be an exhaustion of :0: and let g € H.
Then CAa•xn g} €lN is a Cauchy sequence in I [p;Hk] with limit h. q q ##
It appears that h =Aa g. So Aag E I[p;Hk). Hence I[p ;Hk] c
c I[p;Hk] as sets. From Theorem 2.14 it follows that pis #-sym
metrie.
The following theorem deals with the (semi-} Montel property. Cf. Theo
rem 2.26.
3. 3. Theorem
Let p cw+(Iî) be a moulding set.
The following statements are equivalent:
a. The set p is #-synnnetric and vm€Iî : dim(Hm) < ""·
b. The space 1 [p;~] is semi-Montel.
c. The space Î[p;Hk] is Montel.
d. The space P[p;Hk] is Montel.
Proof:
# a .... b. By Theorem 3.1.d. I[p;Hk] = P[p ;Hk]. Because of Theorem 2.26
the space PCP';Hk] is semi-Montel.
b .... a. Fix m € ll. Let B be the closed 1.mit ball in H • Then B is a m m m closed and bounded set in I[p;Hk]. Hence Bm is compact in Hm' So
dim(ff ) < 00•
m ## Let a € p , let (Irq) qElN be an exhaustion of JI and let g € H. Let
W denote the closure of {Aa·xJI g 1 q € lN} in I[p;Hk]. The set W .. q
c
III Inductive and projective limits
is bounded and closed. Because <Aa•xnq g) qElN is a Cauchy sequence
in W. the element Aa g belongs to (tl c 1 [p;~]· Hence I [p##;Hk] c:
c I[p;Hk] as sets. From Theorem 2.14 it follows that pis #-sym
metrie.
a. * c. This equivalence follows easily from a, * b. and from Theorem
3.2.
a. * d. This equivalence fellows from Theorems 2.26 and 3.2.
The last theorem deals with nuelearity.
3. ll. Theorem
Let pc w+(Il) denote a moulding set, whieh is #-symmetrie.
Then I[p;Hk] is nuelear iff P[p;Hk] is nuelear.
Proof:
Sinee p is #-symmetrie, P[p;Hk] "' 1 [p# ;Hk]. The remaining part of the
103
Cl
proof consists of an application of Theorem 2.16. c
Appendix
In this appendix we want to indicate how the theories [G] and [EGK] fit
in our theory. The spaces introduced in both these papers will he de
seribed in terms of the theory as developed here.
In [G] the spaees SH,A and TH,A are introduced, A is an unbounded,
nonnegative self-adjoint operator in a Hilhert space H. The analyticity space SH,A is defined by
lim ind e -tA(H) t > 0
where e-tA(H) is regarded as a Hilbert space with inner product -tA tA tA
(e h,e g)H, h,g E e (H).
The trajectory space TH;A is the space of all mappings F: (O,oo} -* H, whieh satisfy
(A.1) Vt,t>O e-tA F(t) "'F(t+ T) ,
104 III Inductive and projective limits
endowed with the topology generated by the seminorms F >+- llF(t)llW
F € TH A' where t > O. • +
The sets 'PG and 'PG of functions on 1R are defined by
-tÀ 'Î!G = {À >+- e • À € 1R 1 t > O}
and
'Î!+ is the set of all nonnegative Borel functions f on lR, G
such that for all <.P € \t>G sup <,p(À) f(À) < "'. À€1R
Central results in (G] are:
+ (A.2) The seminoms x i+ llf(A)xllH' x € SH,A' f € \t>G' generate the
inductive limit topology. Cf. [G], Part B, The0rem 1.4.
(A.3) For each F € TH A there exist f € IP~ and b E H sucb tbat -tA ' F(t) = f(A)e h, t > O. Cf. [G], Part B, Tbeorem 2.3.
Now we give a description of the spaces SH,A and TH,A in our terminology.
Let E denote the spectral resolution of the self-adjoint operator A.
Put lI = {m € JN+ 1 E((m-1,m]) /< O}. For each m € IL let Pm denote the
orthogonal projection E((m-1,m]) and let H denote the Hilbert space m
Pm H with induced inner product. So H = em.Ell: Hm. + For t > 0 we define the sequence et E w (:n:) by et(m) -tm
e · m € rr. Put Pe = {et t > O}.
(A.4) The set Pe is a Kötbe set and even a moulding set, because the -1 sequence m o+ (mt + 1) , m E lI, is a Pe-multiplier in R-1(lI).
(A.5) Since Pe,..., {e(l/n) 1 n € JN}, the set Peis type II and #-sym
metrie. Cf. Theorem n1:1. 8.
Fix t > 0. We have
(A.6) v v me:rr (m-1,m]
These inequalities imply that e-tA(H) = H[et]. Cf. Definition 2.6.
So, in the terminology of this cbapter
III Inductive and projective limits 105
Let m € lI and F E Ttt,A• Then, for all t > 0, etA Pm(F(t))
due to the semigroup property (A.1).
eA P (F(1)) 111
The inequalities in (A.4) imply that the mapping F 1+ (eA P111
(F(1)))m.EJ[
is a homeomorphism from T H,A onto P[p; (ff111
)111
€1[] •
We remark that (A.2) corresponds to Corollary 2.13 and (A.3) is a
weaker version of Theorem 2.23.
Because of the properties (A.4) and (A.5) of the set pe• all results in
[G] can be derived from the theory in Sections 2 and 3. An elaborated
and extended version of the'ideas in [G] can be found in (EG 1].
The theory in [G] has been developed into the theory presented in [EGK].
We give a sketch of the spaces S~(A) and T~(A) introduced in the latter
paper. For this, two ingredients are needed:
(A.7) The first ingredient is a set ~. which consists of nonnegative
Borel functions on IRn, bounded on bounded sets and which has the fol
lowing properties
0. The set ~ is quasi-directed, i.e.
1. For each qi E ~the function qi-l defined by
-1 -1 f (q>(À)) '
cp (;\) "' 0 ,
is bounded on bounded sets.
qi(J..) "" 0
cp(À) = 0
2. The supports ~ = {À € IRn ! q>(À) ~ O} of <P, cp E ~. cover the whole
IRn: IRn= U q>. <PE~ -
3. <P(À) :ll y sup ÀtQ(m)
1/J(À) •
106 III Inductive and projective limits
(A.8) The second ingredient is a collection of n strongly commuting
self-adjoint operators, A1, ••• ,A, on a Hilbert space H. It means tbat . n the corresponding spectral projections of the operators Ait• 1 ~ k ~ n,
mutually commute.
Now, let ek denote the spectral resolution of the self-adjoint operator
Ait• k = 1, ••. ,n. Let e denote the joint spectral resolution correspond
ing to the n-set A1, ••• ,An; we mean that eis a projection valued
measure on the o-algebra B(IRn) of Borel sets in IRn, satisfying for all
Borel sets fk in IR, 1 ~ k ~ n,
Let Bb(IRn) denote the set of all bounded Borel sets in IRn.
All spaces introduced in [EGK] are contained in the so-called 'spectra!
trajectory space' G+ which consists of all mappings F: Bb(IRn) ~ H with
the property: F(A1
n A2
) = e(A2
)F(A1), A
1,A
2 E Bb(IRn).
The Hilbert space emb(H) c: G+ is the space of all spectra! trajectories
Fh: A 1+ e(A)h, h E H, endowed with the inner product (Fh,Fg)emb(H) =
= (h,g)H. Naturally, emb(H) and H may be identified.
For q> € ~ the Hilbert space q>(A) • H is the space of all trajectories
F h: A 1+q>(A)e(A n q>)h, h € H, endowed with the inner product q>, . -(Fq>,h'F<P,g)q>(A)•H = (e(~)h,g)H.
Due to the properties of ~. the collection of Hilbert spaces
{q>(A)•ff 1 q> € ~} is an inductive system.
The inductive limit S~(A) c: G+ is defined by
S~(A) = lim ind q>(A)•H • q>E~
+ The space T~(A) c: G is defined to be the space
{F € G+ 1 q>(A)F € emb(H)} ,
III Inductive and projective limits 107
endowed with the topology generated by the seminorms F i+ ll1P(A)Fllemb(H)'
F € T <fî(A) , where IP € <I>.
We remark that the spaces SH,A and TH,A can be identified with the
spaces S<I> (A) and T<I> (A)' respectively. G G +
To the set <I> there is associated a set <I> •
(A.9) The set <I>+ consists of all nonnegative Borel functions f which
satisfy:
- The function f- 1 is bounded on bounded sets,
- Y E<I> : sup f (À) (fl(À) < '" • IP À€lRn
The set <I>+ satisfies the properties O. to 3. in (A.7). Cf. (EGK], Lemma
1 .5.
In [EGK] important results are the following:
+ (A.10) The seminorms F t+ llf(A)Fllemb(H)' F E S<I>(A)' where f € <I> , gener-
ate the inductive limit topology. Cf. [EGK], Theorem 1.7.
(A.11) A set V c T<I>(A) is bounded iff there exists f E <I>+ and a bounded
,set Bin emb(H), such that V • f(A)B. Cf. [EGK], Theorem 2.4.II.
In [EGK] a symmetry condition bas been introduced (axiom A.IV.).
(A.12) A set <I> satisfies axiom A.IV. iff for each IP E <I>++ there exist
Ijl E <I> and y > 0 such that YÀElRn : 1P(À) :;; ylji(À).
We mention an important res'ult:
(A.13) If the set <I> satisfies axiom A.IV., then we have the following
equalities as topological vector spaces:
We now present a description of the spaces S<I>(A) and T<I>(A) in our
terminology.
108 III Inductive and projective limits
Let 1l = {m € 71.n 1 t(C\n) f: O}.
For m € 1l let Hm denote the Hilbert space t(C\n)ff,
The mapping F i+ (F(Q(m)) )m€lI is a bijection from G+ on to mh ffm.
For q> € 41 the sequence d € ut (lI) is defined by d (m) = sup q>Ü,), IP IP À€Q(m)
m € ll. Let p41 = {dq> 1 q> € 41} •
(A. 14) The set p41
c:: 1./ (lI) is a moulding set. The sequence <; € R.1
(lI)
with ë,;(m) = (1 + lml1)-(n+1), m € lI, is a p
41-multiplier.
(A.16) If 41 satisfies axiom A.IV., then the set p41
is #-symmetrie.
Translated in our terminology, we conclude that
The result (A.10) on the inductive limit topology of S4l(A) corresponds
to Corollary 2.13 and the result (A.11) on the bounded sets in T4l(A)
corresponds to Theorem 2.23. The result (A.13) bas been a source of
inspiration for Section 3.
We finish with a technical remark and a genera! remark.
(A.17) The set p41
is #-symmetrie iff for all q> E 41++ there exist y > 0
and 1fi E 41 such that
VÀE U Q(m) : q>(À) ~ Yifi(À) • mE:a:
In fact, the condition that p41 is #-symmetrie is weaker than axiom A.IV.
(A.18) Our theory on the spaces I[p;(ffm)mElI] and P[p;(ffm)m€lI] for
moulding sets p can also be described in the terminology of the theory
in [EGK]. This description is highly nonunique, because there is free
dom in the choice of 41 and of A1, ••• ,An.
CHAPTER IV
SPACES OF ANALYTIC FUNCTIONS ON SEQUENCE SPACES
Introduction
The theory presented in Chapter II and in Chapter III culminates in
this final chapter. With the aid of reproducing kemel theory we intro
duce spaces of analytic functions on sequence spaces, which admit an
I-space structure or a P-space structure. Most ideas and tecbniques in
this final chapter appear to be new.
Let us first describe the sequence spaces, on which the analytic func
tions are defined. For a countable set ID each a € w+ (ID) fixes the
Hilbert space R.2[a;ID] and the semi-inner product space J1,;[a;ID]. Here
R.2
[a;ID] is the space a • R.2 (ID) endowed with the inner product -1 -1 + -1 (x,y) 1+ (a •x,a •y) 2 , and J1,2 [a;ID] is the space a • J1,
2[ID] + Cla • w(ID)
endowed with the semi-inner product (x,y) ,... (a•x,a•y)2
• Clearly, for a
Köthe set g c: w+(ID) the Hilbert spaces J1,2
[a;ID], a Eg, establish an
inductive system generating
Gind[g] "' lim ind J1,2[a;ID] a E g
+ and the spaces J1,2[a;ID], a Eg, establish a projective system generating
Gproj[g) lim proj J1,;[a;ID] • a E g
These spaces have been introduced in [E 2] and fit in the scheme of
Chapter III.
Now we want to introduce function spaces on G. d[g] and G .[g]. Our l.ll prOJ
candidates are:
Fproj [g] = n {F ! F1J1,z[a;ID) E S(J!,2 [a;ID])} , aEg
= u {F 1 F E ea (s<R-2(ID)))}. aEg
110 IV Spaces of analytic functions
We remark that Sa F = F(a•.) and that S(R,;[a;ID]) is defined to be
ea (s <R.2 (ID))). Anybow, F . [g] and F. d [g] are subspaces of A(<,p(ID)). proJ in
The main part of this chapter is devoted to a thorough introduction of
the above spaces F .[g] and Fi.nd[g]. Remarkably, the structure of prOJ these spaces turns out to be very similar to the structure of the
spaces introduced in Chapter III.
Let us explain this a bit. Each a E w+(ID) induces two subspaces of
A(<,p(ID)). The space Find[a] = H(<,p(ID),Ka) with Ka(x,y) = exp(a•x,a•y) 2 and the space F . [a] consisting of all F E A(<,p(ID)) with prOJ ea F E H(<,p(ID),KU). F. d[a] and F .[a] are called elementary spaces. in proJ Next we consider inductive and projective systems, induced by these
elementary spaces. Therefore we involve Köthe sets g, which are conic,
and introduce the compound spaces F. d[g] and F .[g]: in proJ
lim ind F. d[a] aEg rn
lim proj F .[a]. a E g proJ
Initially, the elements of F. d[g] and F .[g] are analytic on <,p(ID). in prOJ However, they admit a larger domain. Indeed, since each element F in
Find[a] bas a unique extension F E ea(S(J1,2(ID))) to Jl,;{a;ID], Find[g]
can be considered as a space of analytic functions on G .[g]. prOJ Further, each element Fin F .[a] uniquely determines an analytic
A proJ function F. in S(J1.2 [a;ID]). So the space F .[g] can be regarded as s
prOJ space of functions on G. d[g]. These functions are ray-analytic. Howin ever, only for type II sets g we are able to prove that they are ana-
lytic on G. d[g]. in
As expected, the compouQd spaces F. d[g] and F .[g] have a very rich in proJ structure:
To begin with, we obtain spaces of analytic functions on <,p(ID) with a
well specified growth behaviour and surprisingly, all are subalgebras
of A(<.P(ID)).
As L.C. spaces F. d[g] and F .[g] fit in the scheme of the previous in prOJ chapter. The elements of F. d[g] and F .[g] are uniquely fixed by . in prOJ sequences in w(:M(ID)) (lM(ID) is the set of multi-indices over ID).
For this we look at the coefficients, labeled by :M(ID), in the monomial
IV Spaces of analytic functions 111
expansions. Introducing a so-called up operation on the collection of + + +
subsets of w (ID), which links to each g c w (ID) a set up [g] c w (:M(ID)),
we arrive at the following identifications:
Fproj [g] - G . [up[g]] • prOJ
So the sequence space Gind[up[g]] induces a function space on the
sequence space G .[g]; similarly, G .[up[g]] induces a function proJ proJ
space on G. d[g]. 1n
As already observed, the spaces G. d[g] and G .[g] fit in the scheme 1n proJ of Chapter III. The sequence space analogue of the important notion of
moulding set from the previous chapter is the notion of generating set,
leading to a very rich topological structure of the spaces Gind[g] and
G .[g]. If gis a generating cone, then up[g] is a generating set and proJ
the spaces F1.nf[g] and F .[g] have a rich topological structure, too. prOJ
We finish with a short description of the contents of this chapter.
Section 1 deals with generating sets and Section 2.with the sequence
spaces they induce.
Section 3 may be considered as an intermezzo. We complete the descrip
tion of the analytic function spaces A(qi(ID)) and A(w(ID)) and describe
their duals in terms of analytic funcions of exponential growth.
Sections 4 and 5 are devoted to F. d and F .• In Section 4 we present 1n prOJ the elementary spaces F. d[a] and F .[a] and in Section 5 the c01D-
1n · proJ pound spaces F. d [g l and F . [g]. In both sec tions reproducing kemel 1n proJ theory is an essential tool.
As an illustration of the theory developed in this chapter, we describe
representations of the generalized Heisenberg groups
in the spaces F. d[g] and F j [g], respectively (Appendix). 1n pro:
112 IV Spaces of analytic functions
§ 1. Generating sets
In Chapter III, Section 1, we have introduced Köthe sets and moulding
sets. A set pc w+(ll), where JI is a countable set, is a Köthe set if
pis separating and quasi-directed. A Köthe set p.c w+(JI) is a moulding
set if there exists <; € R,~ (E) such that <: • p ,..., p.
In this section we consider generating sets. The notion of generating
set is an extension of the notion of moulding set. A set g c w+(ID),
where lD is a countable set, is a generating set if it is a mixture of
a block structure and a moulding set. This means that there exists a
sequence w € w+(lD) without zeroes, a partition {Q(m) 1 m € ll} of ID
and a moulding set p c w+ (ID) such that
In Section 5 of this chapter we introduce inductive limits F. d[g] and in projective limits Fproj[g] of analytic function spaces with the aid of
conic Köthe sets g c w+(ID). If g is a generating cone, then the topo
logical structure of the spaces F. d[g] and F .[g] is similar to the in proJ topological structure of the inductive limits and projective limits,
induced by a moulding set as in Chapter III.
We start with positive sequence sets with a block structure. In the
sequel ID and :n: denote countable sets.
1. 1 • Definitlon
Let Q(Jî) • {Q(m) 1 m € Jî} be a partition of ID and let p be a subset
of w+ (Jî).
The set p (!) Q(E) in w+ (ID) is defined by
P 0 Q(ll) = ~n c(m)xQ(m) 1 c E p}
So the set p 0 Q(ll) consists of sequences, which are constant on the
blocks Q(m), m €ll. Naturally, properties of the set p carry over to
the set p e Q(Il).
IV Spaces of analytic functions
1.2. Lemma
Let Q(][) be a partition of ID and let p,cr c w+(][) be Köthe sets.
Because of Lemma 1.11, up[p] is a moulding set, whence up[g] is a
generating set with skeleton (up[w] ,up[p],R(JM(lI))). o
Finally, we say something about #-symmetry and the up operation.
In Chapter III, Section 1, the positive sequence sets have been classi
fied into three types. The up operation does not disturb this classifi
cation too much.
1. 13. Lemma + Let g c w (ID) be a conic Köthe set. Then
a, The set g is type I or type II iff the set up[9] is type II.
b. The set g is type III iff the set up[g] is type III.
120 IV Spaces of analytic functions
Proof:
a. Let g be type I or type II. Then there exists a sequence (cn)n€lN in
g with en ~ cn+l and with the property that for each a € g there
exists n € lN with a ~ en.
For each n € lN put dn = up[cn]· Then up[g] ,...., {dn 1 n € JN}. We
prove that up[g] is not type I.
Assume up[g] ,.., {dn 1 1 ~ n ~ l} for some l € JN. Then there exists
µ > 0 such that up[2ct] ~ µ up[cil· For all j € ID and p€ lN we have
up[2ct](pej) ~ µ up[ct](pej) or equivalently zP[ct(j))P ~ µ[ct(j)]P.
So for all j € ID we find c t (j) " 0, which is absurd. Hence up [ g] is
type II.
Conversely, let up[g] be type II. Then we can choose a sequence
(cn)n€lN in g, such that up[g] "' {up[Cn] 1 n E lN}. Property (1.5)
implies that g "' {c 1 n E JN}. So g is type I or type II. n
b. This statement is a corollary to statement a.
1. 14. Corollary + Let g c: w (ID) be a type I or type II conic Köthe set.
Then the set up[g) is #-symmetrie.
Proof:
By Lemma 1.13 the set up[g] is type II. Lemma 1.9 yields that up[g] is
separating and directed. Because of Theorem III.1.8 the set up[g] is
c
#-symmetrie. c
EXAMPLE: The set IP+ (ID) is a type II, conic Köthe set. So up[lj)+ (ID)] is
a #-symmetrie Köthe set.
§ 2. Sequence spaces
Tbis section deals with inductive limits G. d[g] and projective limits i.n
G r .[g], subspaces of w(ID), induced by a Köthe set gor a generating p~ .
set g c w+(ID). The inductive limit and the projective limit of function
spaces, find{g] and fproj[g], which will be introduced in Section 5, can be considered as (ray-) analytic function spaces on these G .[g] and prOJ Gind[g], re~pectively. For generating sets g it will appear that Find[g]
IV Spaces of analytic functions
and f .[gJ inherit topological properties of G .[g] and G. d[g]. proJ proJ in respectively.
This section is devoted to the construction of G. d[g] and G .[g] in proJ and some topological properties of these spaces. The spaces G. d[g]
i.n .
121
and G .[g] are sequence spaces. So we give some relations between our prOJ theory of inductive and projective limits and the theory of sequence
spaces.
First, we introduce the notion of sequence spaces.
2. 1. Definition
A subspace Vof w(ID), which is endowed with a L.C. topology, is a
sequence space if for each j E ID the evaluation v 1+ v (j), V E V. is
continuous.
We shall introduce sequence spaces G. d[g] and G .[g] by means of the in pro] construction of the inductive limits and projective limits of Section
111.2. These sequence spaces G. d[g] and G .[g] can also be described l.n prOJ
as inductive limits and projective limits of subspaces in w(ID), which
will be presented now,
2. 2. Definition
Letafw+(ID).
The Hilbert space .Q,2 [a;ID] is defined to be the space a • .Q,2 (ID), endowed
with the inner product
(x.y) .Q,2[a;ID] -1 -1 (a ·x. a •y) 2
The semi-inner product space i;[a;ID] is defined to be the space
The inductive limit G. d[g] and the projective limit G .[g] are in proJ defined by
122 IV Spaces of analytic functions
and
Cf. Definitions III.2.8 and III.2.t9.
We mention the following properties of these limits:
2. 14. Lemma + Let g,h c: w (ID) be Köthe sets with g ""h. Then
G. d[g] = G. d[h] 1n i.n
and
Prooi:
Apply Lemmas III.2.9 and III.2.20.
2.5. Lemma + Let g c w (ID) be a Köthe set. Then
and
Proof:
lim ind Jl.2
[a;ID] a e: g
+ G • [g] = lim proj Jl.
2[a;ID] •
pro1 a e: 9
Let b € w+(ID). The space H[b] used in Definition III.2.6 equals
0
Jl.2[b;ID]. The space H+[b] used in Definition III.2. t7 equals Jl.;[b;ID]. a
(2.1) The spaces G. d[gJ and G .[g] with the pairing in proJ
<x,y> == l. x(j) y(j) • j€ID
(x,y) € G. d[g] x G .[g] i.n proJ
constitute a Hilbertian dual system. Cf. Definition III.2.27.
IV Spaces of analytic functions 123
2. 6. Lemma + Let g c w (ID) be a Köthe set.
The spaces G. d[g] and G .[g] are sequence spaces. 1n proJ
Proof:
We restrict ourselves to. the first case. Let j E. ID. Since e. E. G . [g], J prOJ
we conclude that x 1+ x(j) = <x,ej>' x E Gind[g], is continuous because
of Theorem III.2.29.
Next, we study the spaces G. d[g] and G .[g] where g c:w+(ID) is a 1n pro3
generating set. All results are based on the following theorem.
2 • 7. Theorem + Let g c w (ID) be a generating set with skeleton (W,p,Q(:U:)). Then
and
Proof:
By assumption, g "'W • (p 0 Q(n:)). Lemma 2.4 implies that
and
We use the notations from Chapter II!, S.ec tion 2. For m € n:, let
Hm = i 2 [w·xQ(m)'ID] and then H = Q,2 [w;ID].
Fix c E. w+(l[). The space H[c] introduced in Definition III.2.6 equals
i 2(lxn€n C(m)W • XQ(m) ;ID] and the space H+(c], introduced in Definition
III.2.17 equals i;c~][ c(m)w • XQ(m>'ID].
By Lemma 2.5 we have
Gind(g] lim ind li(c] l[p; (R,2 [w•xQ(m); ID ])m€1L] c E p
and +-
Gproj[g] lim proj 11 [c] = P[p; U 2[w·xQ(m) ;ID])mE.n:l C E p
0
a
124 IV Spaces of analytic functions
We list some results on these inductive/projective limits. The proofs
are omitted, because with the preparations of Section 1 they are trans
lations of results on the inductive/projective limits introduced in
Section III.2.
(2.2) + Let g c w (ID) be a Köthe set.
a. The space Gind[g] is barreled and bornological (Theorem III.2.10).
b. The space Gproj[g] is complete (Theorem III.2.21).
(2.3) Let g c: w+(ID) be a generating set.
a. The inductive limit topology of G. d[g] equals the topology generated # . 1n
by the seminorms Pa• a Eg , defined by pa(x) = la•xl 2, x € Gind[g].
(Corollary III.2.13.)
b. The space Gind[g] is nuclear iff Va€g VbEg#: a•b E t 1(ID). (Theorem III.2.16.)
c. A bounded (compact) subset W of G • [g] is of the form a • .B where a . prOJ belongs to g# and B is a bounded (compact) subset of t
2(ID).
(Theorem III.2.23 and Corollary III.2.24.)
d. Gproj[g] is semi-Montel iff VaEg Vb€g# : a•b E c0 (JD).
(Theorem III.2.26.)
e. The space Gind[g] is a representation of the strong dual of Gproj[g].
(Theorem III.2.30.)
(2.4) + Let g c: w (lD) be a #-symmetrie generating set.
# # a. Gind[g] = Gproj[g ], Gproj[g] = Gind[g. ],
The spaces G. d[g] and G .[g] are reflexive. · 1n - proJ (Theorem III.3.1.)
b. The space G .[g] is a representation of the strong dual of G1.nd[g]. proJ (Theorem III.2 .• 31.)
c. If VaEg VbEg# : a•b € c0 (ID), then both spaces Gind[g]and Gptoj[g] are Montel.
(Theorem III.3.3.)
IV Spaces of analytic functions 125
d. If VaEg VbEg# : a•b E .R-1 (ID), then both spaces Gind [g] and Gproj [g]
are nuc lear.
(Theorem III.3.4.)
Lemma 2.6 states that G. d[g] and G .[g] are sequence spaces. We . in proJ give some relevant topics of the sequence space theory as founded by
KÖthe. These topics can also be found .in Kamthan and Gupta, [KG],
Chapter II.
Let S c w(ID) be a sequence space.
* The space S is normal if for each u E S the normal cone
{V E w(ID) 1 IVI ~ lul} of u is contained in S. It turns out that
the space S is normal iff .R,00
(ID) • S = S.
* The space Sx c w(ID) is defined by
Sx = {v E w(ID) 1 VUES u•v E i1
(ID)} •
The space Sx has been put forward as a natural dual of S. It is
cal led the Köthe dual or the arass of S. We have S c Sxx and
Sx = Sxxx. The space Sx is normal.
* The space Sis a Köthe spaae if S = Sxx. So Sx is a Köthe space.
* For each v E Sx the seminorm !tv on S is defined by !t/U) = lu~!v 11
,
U ES. The seminorms !tv' V E Sx, generate the normal or Köthe topo
logy of S.
In the following lemma we connect facts on Köthe duals with our results
on inductive/projective limits. • x +-
First of all: (.R-2
[a;ID]) = .R-2
[a;ID].
2.8. Lemma + Let g c w (ID) be a generating set. Then as sets
a.
b.
# = G. d [g ] in:
G [ #] = G. [g##] proj g ind
126
c.
Proof:
G. [g##] ind
IV Spaces of analytic functions
-1 Let g have a skeleton (w,p.Q(lI)). Then w • g is a root set, i.e. a
generating set with weight sequence n. Further we have
and
-1 =w•G.d[W •g] in
-1 -1 Gproj[g] = w • Gproj[W • g] •
The remaining part of the proof consists of an application of [E 2], -1 -r Proposition 8.12, to G. d[w • g] and G • [w • g]. o in proJ
So the space G .[g] is always a Köthe space. The space G;nd[g] is a prOJ "
Köthe space iff the set g is #-symmetrie. This shows again the relevance
of the #-symmetry condition.
2.9. Lemma + Let g c: w (lD) be a generating set with skeleton (w,p,Q(lI)). Let
è; E R-1 (:JI.) denote a p-multiplier, such that ~Il ~(m)XQ(m) E R-1 (D).
(This condition implies that the spaces G. d[g] and G .[g] are in proJ nuclear.) Then
a. The inducti.ve limit topology of G. d[g] is equal to the normal topol.n
b. The projective limit topology of G .[g] is equal to the normal topo-prOJ logy of G .[g]. prOJ
Proof:
Just like in Lemma 2.8 we must replace the generating set g by the root -1 set W • g. Further, see [E 2], Lemma 8.16 and Corollary 8.17. o
The Lemmas 2.8 and 2.9 show that there is an overlap between our theory
of inductive/projective limits and the Köthe space theory.
IV Spaces of analytic functions 127
§ 3. A sequence space representation of A(q>(ID)) and A(w(ID))
In this section we introduce an injective mapping Seq from A(q>(ID)) into
w(JM(ID)). After some preparation in Section 4. in Section 5 we introduce
inductive limits Find[g] and projective limits Fproj[g] of analytic
functions. which belong to A(qi(ID)). So, having defined the mapping Seq
in A(<.p(ID)), the mapping Seq yields sequence space representations of
F. d[g] and F .[g]. These representations deepen our insights in the in proJ topological structure of the spaces f. d[g] and F .[g]. in proJ
This section is divided into two parts. The topics in the first part
are the sequence space representation of A(q>(ID)) and A(w(ID)) induced
by the mapping Seq. The L.C. spaces q>(ID) and w(ID) are nuclear. In
literature one can find extensive discussions on spa.ces consisting of
analytic functions on nuclear L.C. spaces. Cf. [Di 1] and the references
therein. We present a sequence space representation of A(q>(ID)) and of
A(w(ID)) as a demonstration of our techniques. The second part of this
section is devoted to the strong duals of A(q>(ID)) and A(w(ID)) and is
not connected with the remaining part of this chapter.
We introduce the mapping Seq. To this end we need the differential
operator;; 35 , s E :M(ID), defined on A (qi(ID)). Cf. (4. 6) in Section I. 4.
3. 1 • Definition For each F E A(<t>(ID)) the s.equence Seq[F] E w(lM(ID)) is defined by
Seq[F](S) "-1- [as F] (lfl) • Tsï
S € lM(ID) •
We give some properties of the operator Seq, in which the following
functions occur:
- The functions <lis, s € :M(ID), on<.p(ID) with cl>5
(x) --1-x5•
rsr - The function K on <.p(ID) x <.p(ID) with K(x,y) • exp(x,y) 2•
(3.1) Each F € A(q>(ID)) satisfies
v E (ID): F(X) = i Seq[F](S)<lis(X) X !.P SE:M(ID)
128 IV Spaces of analytic functions
where the series converges absolutely and uniformly in x on compact
sets in <P(lD). Cf. Theorem I.4.19.
(3.2) LetFEA(q>(ID)) and let a€w+(ID).
Then the function 0a F with [0a F](x) = F(a•x) belongs to A((jl(ID)) and
Seq [0a F] = up[a] • Seq[F].
(3.3) The spac.e q>(lD) is dense in 4!R.2 , whence F i+ Flq>(ID)' F E S(.R.2 (ID))
is a unitary operator from S(t2(lD)) onto H(1P(lD),K). Cf. Theorem
II.1.14. The results on S(t2(1D)) in Section II.2 imply that
H(1&>(ID),K) c: A(cp(ID)) and that H5
1 s E lM(ID)} is an orthonormal basis
in H(q>(ID) ,K). So for F € H(qi(ID) ,K} and s E lM(lD) we have
Seq[FJ(s) = (F ,'1\)H(\&)(ID) ,K) •
Property (3.3) leads to:
3.2. Lemma
The operator Seq is a unitary operator from H(q>(ID) ,K) on to t2
(lM(ID)).
Proof:
The proof is based on the relation Seq[F](S)
Next, we introduce the ~nnounced sequence space representations of
A(q>(ID)} and A(w(ID)). It turns outt that the ranges of A\(1P(lD)) and
A(w(ID)) under the operator Seq are as sets equal to projective limits
of the type G .[9] with ga generating set of the special type intro-prOJ
duced now.
3. 3. Definition
Let a E w+(ID) and g c w""(m).
The sequence v E w(lM(ID)) is defined by
V (S ) " 1 / ISf .
The sequence ev[a] E w+(lM(lD)) and the sequence set ev[g] c w+(lM(ID))
are defined by
ev[a] = v • up(a] and ev[g] = v • up[g] •
c
IV Spaces of analytic functions
REMARK: + For all a ,b E. w (ID) we have up [a] • ev [b]
3.ll. Lemma + Let g c w (ID) be a generating cone.
ev[a•b]
Then the set ev[g] is a generating set in w+(lM(ID)).
Proof:
Theorem 1.12 implies that up[g] is a generating set in w+(ll:(ID)).
Since V = (1//ST)SElM(ID) has full support, we get by Lemma 1.7 that
129
ev [g] v • up [g] is a generating set. a
So, ev[cp+(ID)] and ev[w+(ID)] are generating sets.
Before we state the theorem on the sequence space representations of
A(cp(ID)) and A(w(ID)), induced by Seq, we recapitulate some relevant
results from Section I.4.
(3. 4) A complex valued function F on cp(ID) is analytic on q>(ID) iff F
is ray-analytic on cp(ID).
(3.5) A co!ltplex valued function G on w(ID) is analytic on w(ID) iff G
is ray-analytic on w(ID) and there exists a finite subset 1F c ID such
that for all x E w(ID) G(x) = G(XlF • x).
(3.6) The mapping Go+ Glcp(ID) is an injection from A(w(ID)) into
A(cp(ID)). We identify the function G E A(w(ID)) with its restriction to
cp(ID). So A(w(ID)) c A(cp(ID)).
We come to the main result of this section.
3. 5. Theorem
a. The operator Seq maps A(cp(ID)) bijectively onto G • [ev [q>+ (ID)]]. prOJ
b. The operator Seq maps A(w(ID)) bijectively onto G .[ev[w+(ID)]]. prOJ
Proof:
a. Fix F E A(cp(ID)). For all X E q>(ID) we have
F(X) = l Seq[F](S) • 4> 5 (-X) = l (Seq[F] • ev[X]) (S) • sElM(ID) SElM(ID)
130 IV Spaces of analytic functions
From (3.1) it follows that Seq[F] • ev[a] E R.2 (lM(ID)) fo;r each
a E q:,+(ID), whence Seq[F] EG .[ev[q>+(ID)]]. proJ +
Conversely, let f EG .[ev[q>+(ID)]]. Let C E R.1
(ID) denote a 1.fJ+(ID)-proJ +
multiplier with ICl1 < 1. Then up[?;] E R.1 (lf(ID)) by Theorem 1.10.
Let a E q>+ (ID). Since C bas full support, we have
-1 f • ev[a] = up[CJ • f • ev[( • a] •
-1 + -1 Because ( • a E <P (ID), the sequence f • ev[C • a] belongs to
R.00
(:M(ID)), whence f • ev[a) E ~(M(ID)).
The series lsE:JM(lD) (f • ev[x]) (S) is absolutely and unifonnly con
vergent in x on compact sets in q.i(ID), so the complex valued function
F on q>(ID), defined by F(x) = lsEM(ID) (f•ev[x])(s), x E ql(ID). is
analytic on all finite dimensional subspaces W of q>(JD). Corollary
I. 4. 16 implies that F E A(l,fJ(ID)). We have Seq [F) = f.
Evidently, the operator Seq is injective.
b. Let GE A(w(ID)). There exists a finite subset IF of ID such that for
all x E w(lD), G(x) = G(xIF• x). This leads to the relation
Seq[G) = Seq[G]. up[xIFJ •
Since fora E w+(ID) the sequence XIF• a belongs to q>+(ID), we have
Seq[G] • up[a] • Seq[G] • up[xIF• a] € R.2
(:M(ID))
+ So, Seq [G] E G . [ev [<P (ID) 1J. · prOJ + +
Conversely, let g EG .[ev[w (ID)]] c: G .[ev[q> (ID}]]. By state-proJ proJ
ment a. of this theorem there exists G € Á(q>(ID)) with Seq[G] = g. We prove that G belongs to A(w(JD)) by proving that there exists a
finite subset IF of ID, such that for all x E w(ID) G(X) = G(XIF• X).
In terms of g = Seq[G] this means that g = g • up[xIF].
So suppose the opposite, i.e. for every finite subset IF of ID we
have g # g • up[XIF).
First note that for all subsets IF of ID and all s E M(JD)
= \ 1 ,
( 0 •
ID[s] c:: IF
ID[S] </:. IF
IV Spaces of analytic functions 131
Since g # Gl, there exists s 1 E 1M(ID) with g (s 1
) # 0. For J/, > 1 we
proceed inductively. We define SJ/, E :M(ID) as fellows;
Since IDJ/,-l = u~:: ID[sk] is a finite subset of ID, there exists
SJ/, E 1M(ID) with the properties
Next we define a sequence b E w+·(ID) and we will prove that
g• èv[b] f. J/,00
(1M(ID)). Thus we arrive at a contradiction. Let (jJ/,)J/,EJN
denote a sequence in ID with jJ/, E ID[SJ/,] and jJ/, '/. IDJ/,-l' The jJ/,'s are
all distinct.
The sequence b E w+(ID) is defined by
b (j) = 1 ' j # j J/, for all J/, E 1N
J/, E 1N •
For all J/, E 1N we have 1 (g • ev[b]) (SJ/,) 1 ~ !/,,
The previous theorem suggests a way to define L.C. topologies in
A((j)(ID)) and in A(w(ID)). In the sequel we consider A(\P(ID)) and
A(w(ID)) as L.C. spaces with the coarsest topology, which makes Seq a
continuous bijection from A(\P(ID)) on to G . [ev [\P +(ID)]] and from + prOJ
A(w(ID)) onto G .[ev[w (ID)]], respectively. We remark that both topo-proJ
logies are finer than the compact open topology, i.e. the topology of
uniform convergence on the compact subsets of (j)(ID) and w(ID), respec
tively.
The final part of this section deals with the strong duals of A((j)(ID))
and A(w(ID)). +
As we have seen, the sequence space G. d[ev[(j) (ID)]] can be regarded as + in
the strong dual of G .[ev[(j) (ID)]]. Cf. Result (2.3). So the natura! proJ
candidate for a representation of the strong dual of A((j)(ID)) is given
by Seq-1 [G. d[ev[(j)+(ID)]]]. Note that G. d[ev[(j)+(ID)]] is contained in in in
G • [ev[w+ (ID)]]. Hence, by means of the mapping Seq, we obtain a prOJ
description of the strong dual of A((j)(ID)) in terms of a subspace of
a
132 IV Spaces of analytic f unctions
A(w(ID)). Similarly, Seq- 1[Gind[ev[w+(ID)]]], a subspace of A(tp(ID)),
is a description of the strong dual of A(w(ID)).
Our results on the strong duals of A(q>(ID)) and A(w(ID)) are natural
infinitely dimensional analogues of a result by J.P. Antoine and
M. Vause [AV]. In [AV] they d iscuss a pairing between two analytic
spaces on t. They deal with the space A(t) consisting of all analytic
functions on t and with the space Exp(t), consisting of all entire
functions F of exponential type, i.e.
See also example 6 in [EGK].
Here we find ourselves in similar. situations. We remarked already that
the strong dual of A(q>(ID)) can be represented by a subspace of A(w(ID)).
3.6. Theorem
Let F E A(w(ID)).
Then Seq[F] EG. d[ev[qt(ID)]] iff in
Proof:
Assume Seq[F] € G. d[ev[q>+(ID)]]. Then there exist a E Q>+(ID) and l.ll
g € t 2 (:M(ID)) such that Seq[F] = ev[a] ··g and for all X E w(ID) we have
IF(x) 1 = 1 I +,- g(s) (a•x) 5 1 ~ sElM(ID) .
~19100 • I ~(a•lxl) 5 = S€1M(ID) S'
lgl00
• exp(<a, lxf>)
Conversely, let a € Q>+ (ID) and y > 0 such that
VxEw(ID) : IF(x) 1 ~ y exp(<a, lxl>) •
. + We show that Seq[F] = ev[b] • g for some b E Q> (ID) and g E i
2C:M(ID)),
With the aid of the Cauchy integral (cf. Lemma I.4.18) we obtain for
IV Spaces of analytic functions 133
all r € w +(ID) with ID[r] • ID and all s E JM(ID} the estimate
(3.7) 1 [ël 5 F](t))I ::> ysl exp(<a,r>)r-s •
Let S E JM(ID).
If there exists j E ID[S] with j </. ID[a] we substitute r = (),-1)e. + n
in equality (3. 7) and we obtain 1 [as F](ll} 1 ::! y s! À-s(j). Taking ÀJ + 00,
it follows that [3 5 F](~} = O.
If ID[S] c: ID[a], we substitute r = s·a-1 +os in inequality (3.7) and
we obtain llosF](ll)I ::! ysl exp(lsl1)s-s as ::!y(ea)5
• So !Seq[FJI ::!>
::! y• ev(ea].
Putµ = 2 • #(ID[a]). Then l 11/vl 2 < 1 and up[11/µ] E J1,2 (:M(ID)).
Put b = µea and g = Seq[F] • ev[b-1], Then b E q>+(ID) and since
lgl ::! Y up[11/lll, g E t2
(JM(ID)). Further we have Seq[F] = ev[b] • g. c
The strong dual of A(w(ID)) can be represented by a subspace of A(q>(ID)).
We have a similar result as in Theorem 3.6:
3. 7. Theorem
Let F E A(q>(ID)).
Then Seq[F] E Gind[ev[w+(ID)]] iff
3bEw+(ID)3y>O '°"xEc.p(ID): IF(X)I ::! y exp(<b,lxl>).
Proof:
The proof is based on similar argtullents as the proof of Theorem 3.6. a
The two previous theorema lead to the introduction of the spaces of
functions of exponential type on w(ID) and c.p(ID), respectively.
3. 8. Definition
The spaces Exp(w(ID)) and Exp(c.p(ID)} are defined by
and
Exp(w(ID)) = Seq-1 [G. d[ev[c.p+(ID)]]] 1n
-1 + Exp(tp(JD)) = Seq [G . [ev[w (ID)]]].
prOJ
134 IV Spaces of analytic functions
The pairing between A(q>(ID)) and Exp(w(ID)) is given by
<F,G> F E A(q>(ID)) • G € Exp(w(ID)) •
The pairing between A(w(ID)) and Exp(q>(ID)) is completely similar.
§ 4. Elementary spaces in A(cp( ID))
+ In Section 2 we have proved that for a Köthe set g c w (ID)
G. d[g] = lim ind fl2[a;JD] and G • [g] = lim proj fl+2[a;ID] • in a € g proJ a E g
This description of the spaces G. d[g] and G .[g] is the guide for J.n prOJ
the construction of our analytic function spaces F. d[g] and F .[g], in pro3 the compound spaces. For each a E w+(ID) we introduce two elementary
spaces, the Hilbert space F. d[a] and the semi-inner product space in f .[a]. These elementary spaces are the building blocks for the compro1 pound spaces F. d(g] and F . [g], viz.
J.n prOJ
F. d[g] = lim ind find[a] rn a E g
and F .[g] = lim proj F .[a]. proJ a E g proJ
This section consists of two similar parts, The first part deals with + the elementary spaces F. d[a], a € w (ID). Amongst others, we study
l.Il
continuous.functions on fl;[a;ID] which are extensions of the functions
in F. d[a] and we study the sequence space representation of F. d[a] in ].Il
induced by the operator Seq. The second part deals with the elementary
spaces F .[a]. Here we discuss similar topics as in the first part. prOJ
The elementary spaces F ind [a J
The spaces F. d[a] are functional Hilbert spaces determined by a special in class of functions of positive type on q>(ID).
ll. 1. Definition
Let a E w+(ID).
The function Ka E PT(q>(ID)) is defined by
IV Spaces of analytic functions 135
x,y € q>(ID) •
The elementary space Find[a] is defined as the functional Hlilbert space
H(q>(ID) ,K3).
We denote K11
by K and F. d[11] by F. in
4.2. Lemma
Let a € w+(ID) and let FE Find[a].
The function 9 1
F belongs to F and a-
lt9 _1 FUF ~ UFHF. [a] • a ind
Proof:
Let V(q>(ID),Ka) denote the linear span <{Ic; 1 xEq>(ID)}>. a ~.Q, a
Let GE V(q>(ID),K ). Then G = lj=l a'j KYj· So,
and
0 _1
G
a
.Q,
L a. Ka•y. j=1 J J
Since V(<.P(ID),K3
) is dense in f. d[a], we find that 0 -l maps F. d[a] in a in isomorphically into F.
4. 3. Corollary +
Let a E w (ID) and F E F. d[a]. in The function Fis analytic on q>(ID).
Proof:
We only have to prove that F is ray-analytic on q>(ID). By Lemma 4.2 the
function 9 -1 F belongs to F. Lemma II.1.14 implies that there exists a a
function G € S(.Q,2(ID)) = H(R-2 (ID),exp( , ) 2 ) such that 8a_1 F = Glq>(ID)'
Since G is analytic on t 2 (ID), the function 9a-1 F is ray-analytic on
0
q>(ID). Hence F itself is ray-analytic on q>(ID). o
136 IV Spaces of analytic functions
The following results justify the use of the elementary spaces Find[a]
as building blocks for inductive limits. Cf. Definition I.2.1.
4. 4. Theorem + Let a,b E w (ID).
Then a :;; b iff F. d[a] c::;.. F. d [b]. in in·
Proof:
Lemma II.1.10 states that
Assume Ka :;; µKb for some µ > 0. Then for all À > 0 and all j E ID
Hence a :;; b.
Conversely, let a :;; b. Since for all u E q>(ID), (a•u,a•u) 2 Lemma II.1.17 implies that Ka:;; Kb.
4. 5. Corollary + Let g c w (ID) denote a conic KÖthe set.
Then {F. d[a] 1 a E g} is an inductive system. in
Proof:
Since g is a cone, the set g is a directed set. For a,b E g with a :;; b,
0
Theorem 4.2 implies F. d[a] c-...F. d[b]. c in in
Each function F E F. d[a] can be extended uniquely to a continuous + in
function on R.2
[a;ID].
4.6. Theorem
Let aE w+(ID) and let F E F. d[a]. in . +
There exists precisely one continuous function G on i 2 [a;ID], such that
F = Glq>(ID)" The function G is ray-analytic on ,Q,;[a;ID].
IV Spaces of artalytic functions
Proof: ·
First we prove that F = 811 a F ~
Let x € cp(ID). Since F is ray-analytic, the func tion f, defined by
f(À) = F(11 3 • x +À Gla • x), À E q:, is entire. Since F € Find [a], there
exists y > 0 such that
Liouville's theorem implies F(x) = f(1) = f(O) = F(11 • x). Hence a F = 8tl F.
a
137
The function Sa-t F belongs to F. There exists H € S(.e,2 (ID)) such that
8 -1 F =Hl (ID)• Cf. the proof of Corollary 4.3. a qi + . +
We define G on t2
[a;ID] by G(x) = H(a•x), x E R.2 [a;ID].
The function G is continuous on .e.;[a;ID] and for all x E qi(ID)
G(x) = H(a•x) = F(na • x) = F(x) •
Since qi(ID) is dense in R.;[a;ID], there exists exactly one such function
G. Remark that G is ray-analytic on .e.;[a;ID]. o
Since the space F. d[a] is contained in A(IQ(ID)), the operator Seq in
induces a sequence space representation of Find[a].
Il. 7. Theorem
Let a E w+(ID).
The operator Seq is an isometrie isomorphism from F. d[a] onto lil
t2
[up[a];:M(ID)].
Proof:
Let F E F. d[a]. Then l.Il
Seq[F] = up[a] • Seq[S _1
F] € R.2
[up(a];:M(ID)] • a
-1 -1 Conversely, let f E R.2 [up[a];:M(ID)]. Put F =Sa (Seq [up[a ] • f]).
-1 Then F E F. d[a] and Seq[F] = up[a] • up[a ] • f = f. in By Lemmas 4.2 and 3.2 we have for all F € F. d[a] in
138 IV Spaces of analytic functions
USeq [F] Il R.2
[up [a] ;1M(1P)] 11Seq[9 _1 F]llR, (1M(ID)) = a 2
119a-1 FllF = llFllF. [a] • ind
The elementary spaces F . [a] prOJ
The spaces F .[a] are semi-inner product spaces defined in the followprOJ
ing way.
4. 8. Definition
Let a € w+(ID).
The elementary space F .[a] is defined as the space prOJ
{F € A(11>(1P)) 1 ea F € F}
endowed with the semi-inner product
Sometimes elementary spaces of bot.h types coincide.
4.9. Lemma + Let a,b E w (ID).
Then a • b = 11 iff Fproj [a] Find [b].
Proof:
Since for all x € q>(ID) and F € F . [a] prOJ
the subspace 911
(F . [a]) of F . [a] is a functional Hilbet't space a prOJ -1 prOJ witb reproducing kemel Ka We have
and
iff a • b = 11 •
0
IJ
IV Spaces of analytic functions 139
In particular, Fproj[n] = Find[n] =F.
The next theorem is the analogue of Theorem 4.4.
4.10. Theorem + Let a,b E w (ID).
Then a:;; b iff Fproj[b] c;..fproj[a].
Proof:
""'·Let F € F .[b]. Then 0bF E f. Since b-1 ·a:;; n, the operator proJ 0 -l 1 F is a bounded diagonal operator on F. So 0 F = 0 _1 (0b F) E f b ·a a b •a
and
". Fix j € ID [a] , For r > 0 we define Gr € A(q>(ID)) by Gr (z) = = exp(h• z2(j)), z E qi(ID). Lemma II.3.10 implies that G € f iff
r 0 < r < 1. So 0a Gr € f iff 0 < r < a-2
(j). Hence 0b Gr f. f for
r ~ a-2(j). We find that b(j) r O and a-2(j) ~ b-2(j). Soa :;; b. a
4. 11. Corollary + Let g c w (ID) denote a conic Köthe set.
Then {f .[a] 1 a Eg} is a projective· system. prOJ
Proof:
Since g is a cone, the set g is directed. For a,b E g with a :;; b,
Theorem4.10impliesF .[b]c .... F .[a]. a proJ proJ
By Theorem 4.6, to each function F € F. d[a] there is associated pre-1n
cisely one continuous function on ~;[a;ID]. To each function F E F .[a] there is associated precisely one analytic prOJ function on ~2 [a;ID].
4. 12. Theorem
Let a E w+(ID) and let F E F .[a]. prOJ There exists precisely one function G E S(t2[a;ID]) such that
140 IV Spaces of analytic functions
Proof: -1 -1
The function Flna•q:>(ID) belongs to H(n8
•q:i(ID);exp[(a •.,a •.)2]).
By Theorem II.1.14 there exists G E s(R,2 [a;ID]) such that
Fin •q:>(ID) =Gin •<tJ(lD}. a a
The function G is analytic on t 2 [a;ID] and since na• q:>(ID) is dense in
t 2 [a;ID], the function G is uniquely determined. c
Finally, we deal with the sequence space representation of the elementary
space F .(a], induced by the operator Seq. prOJ
4.13. Theorem
Let a € w+(ID}.
..... a. The i>perator Seq -maps F . (a] continuously into t
b. c:. Statement a. and Theorem 3.5 imply this inclusion.
=>. Let f € R-+2[up[a];JM(ID)] n G .{ev[q:i+(ID)]]. Put g = f - up[tla] •f. prOJ
By Theorem 3.5 there exist F,G € A(q:i(ID}) such that -1 -1
F = Seq [up[11a]· f] and G = Seq [g].
IV Spaces of analytic functions 141
Then F + G E A(qi(ID)) and 0a (F + G)
and Seq[F +G] = f.
0aFEF.SoF+GEF .[a] · prOJ
§ 5. Compound spaces in A(qi(ID))
In this section we define inductive limits F. d[g] in F .[g] of elementary spaces for conic Köthe sets prOJ
and projective limits + 9 c w (ID). We call
a
these limits F. d[g] and F .[g] càmpound spaces. The elements of these · in pro3 compound spaces are analytic functions on qi(ID).
We mention two important properties of these new spaces:
First, the spaces Seq[F. d[g]] and Seq[F .[g]J are inductive limits in prOJ and projective limits, respectively, of sequence spaces of the kind
introduced in Chapter III. So results of Chapter III apply.
Second, the elements of F. d[g] and F .[g], being analytic functions in proJ on tp(ID), can be extended to analytic functions on G .[g] and to ray
prOJ analytic functions on G. d[g], respectively. in
The subdivision of this section is as follows
The first two parts deal with the compound spaces F. d[g) and the comin pound spaces F . (g], respectively, induced by conic Köthe sets prOJ g c u/ (ID). In the third part we study the compound spaces induced by
generating cones in w+ (ID).
We start with the inductive limits F. d[g]. l.n
5. 1. Definition + .Let g c w (ID) denote a conic Köthe set.
The compound space F. d[g] is defined by l.n
lim ind Find[a] • a E g
+ For each conic Köthe set g c w (ID) we have
+ + F. d[q> (ID)] c;... F. d[g] c;.. F. d[w (ID)] in in in
142 IV Spaces of analytic functions
The space Find[g] is an algebra in the usual way.
5.2. Lemma
Let a,b E t/(ID), let F E F. d[a] and let GE f. d[b]. in in Then F • G E f. d[a+b] and in
IF •Gif. d[a+b] :; DFllF. [a]. llGllF. [b] in ind ind
Proof:
The proof is based on Lemma II.1.7. Put
a= 2 2 RFHF. [a] and 13 = llGllF. [b] •
ind ind
Let t E IN and a.. E 4:, x. E q>(ID), 1 ::> j ::> R.. Then . J J
J/,
y = 1 I a. F(x.}G(x.)1 2 ~ j=1 J J J
Now application of Remark (II.1.4) and Lemma II.1.20 gives
R,
y ::>. a.{3 I ëi"k a. j exp [ ( a • xk, a • x j ) 2 + ( b • xk, b • x j) 2 ] • k,j=1
Since (x,y} * exp[(a•x,b•y) 2 + (b•x,a•y) 2 ] is a function of positive
type, greater than (.i<,y) ~ 1, Lemma II. 1. 20 implies
Hence F • G E f. d[a+b]. By Corollary II.1.8 we get in
llF • Gllfi .. nd[a+b] ;:> llFllF. [a] • llGllF. [b] ind ind
c
IV Spaces of analytic functions
5. 3. Corollary + Let g c w (ID) be a conic Köthe set.
a. For all F,G E F. d[g] the function F •G belongs to F. d[g). in in
b. For all GE F. d[g] the mapping F 1+ F • G is a continuous linear in operator from F. d[g] into F. d[g]. in in
Proof:
143
a. Let a,b E g. Since g is a conic Köthe set, there exists c E g, such
that a+b ~ c. So a product of two elements of Find[g] also is an element of F. d[g]. in .
b. Let G E F ind [b] for some b E g. Let MG denote the operator F i+ F • G.
Lemma 5.2 implies that the operato.r MG maps the space Find [a] con
tinuously into find[g]. Since MG is a linear operator, Theorem I.2.3
yields that MG is a continuous operator on Find[g]. o
We characterize the elements of F. d[g] in terms of a growth estimate. in
5. 4. Theorem + Let g c w (ID) be a conic Köthe set. Let F be a complex valued function
on <P(ID).
Then F belongs to F. d[g] iff there exists a € g and y > 0 such that in
Q,
1 ~ a. F"<X:TI z j=1 J J
Proof:
The statement is equivalent to:
The function F belongs to F. d[g] iff there exists a € g such that in F E F. d[a]. Cf. Theorem II.1.7. o in
The next theorem states that Find[g] can be identified with a subspace
of A(G • [g]). prOJ
144 IV Spaces of analytic functions
5.5. Theorem + Let g c.: w (ID) be a conic Köthe set and let FE Find[g].
There exists a unique element GE A(G .[g]) such that Fis the prOJ restriction of G to qi(ID).
Proof:
There exists a E g such that F E F. d[a]. By Theorem 4.6 it follows 1.n
that F has a unique continuous ray-analytic extension H to !l;[a;ID].
Since G .[g] c;..!l+2 [a;ID], the function G = HIG [ ] is analytic on prOJ , g G .[g] and is uniquely detennined. proJ o
proJ
One of the main results in this section is the fact that the space
find[g] bas the same structure as the inductive limit Gind[up[g]] in
w(:M(ID)).
s. 6. Theorem + Let g c.: w (ID) be a conic KÖthe set.
The linear operator Seq maps Find[g] homeomorphicly onto Gind[up[g]].
Proof:
Lemma 4. 7 implies that the operator Seq maps F. d [a] homeomorphic ly in
onto !l2[up[a];lM(ID)]. Theorem I.2.3 implies that the operator Seq is a
homeomorphism from f. d·[g] onto in
lim ind !l2 [up[a];lM(ID)] = Gind[up[g]] • a Eg
If g is a conic Köthe set, then the set up[g] is a Köthe set, so the
topological properties of F. d[g] which are summarized at the end of in
this section, follow immediately from Section 2.
In the next part we consider the projective limits F .[g]. proJ
5. 7. Definition + Let g c.: w (ID) be a conic Köthe set.
The compound space F .[g] is defined by proJ
fproJ.[g] = lim proj F .[a] • a E g proJ
0
IV Spaces of analytic functions 145
For each conic Köthe set g c w+ (ID) we have
We charactèrize the elements of F .(g] in terms of a growth estimate. proJ
s. 8. Theorem + Let g c: w (ID) be a conic Köthe set and let F denote a complex valued
function on cp(ID).
The function F belongs to F .[g] iff for alla€ g there exists y > 0 prOJ such that
v v R.EIN a.E4;,x.Ecp(ID), 1 J J
Proof:
". Since F € Fproj[g], we get that for alla€ g the function F belongs
to F . [a]. Hence, for all a E g, the function 0a F belongs to F. prOJ Apply Lemma II.1.7.
*'•Lemma II.1.7 implies that the function x 1+ F(a•x), x E cp(ID}, belongs
to F. If F is analytic on cp(ID), then F E F • [a] for all a E g. So prOJ
we only have to prove that F is analytic on cp(ID) or equivalently
that Fis ray-analytic on cp(ID). Cf. Theorem I.4~15.
Let u,v E cp(ID). Since g is a Köthe set, there exists a Eg such that
the supports ID[u] and ID[V] are contained in ID[a]. The function
À E C
is entire, 0
Just like the compound space F. d [g], the compound space F . [g] is an in proJ
algebra in the usual way.
146 IV Spaces of analytic functions
5.9. Theorem + Let g c w (ID) be a conic KÖthe set. Let F,G € F . [g].
proJ Then the function F • G belongs to F • [g] and satisfies for all a € g
prOJ
Proof:
Let a € g. Put a = 110rzaFllF and 8 = 110v'2aGllF. We apply Lennna II.1.7.
Let R, € lN, aj € 4:, Xj € q>(ID), 1 ;;> j ;;> R,. Then
So 0a (F • G) belongs to F and by Corollary II. 1. 8
Theorem 5.8 implies that the function F • G belongs to F • [g]. prOJ
Tbeorem 5.9 states tbat the product in F .[g] is jointly continuous, prOJ
in contrast with Corollary 5.3, which states tbat the product in F. d[g] in
is separately continuous.
c
IV Spaces of analytic functions
Under the condition tbat g is type II, 'We will prove the analogue of
Theorem 5.5, which reads: Each F € F . [9] has a unique analytic prOJ
extension G on G. d[g]. For arbitrary conic power sets g we have the l.n
following weaker result:
5.1 O. Theorem + Let g c: w (ID) be a conic Köthe set. Let F € F . [g].
prOJ Then the function F bas a unique ray-analytic extension G on G. d[g],
l.n
147
such that for alla€ g the restriction Gl.e,2
[a;ID] belongs to sU2 [a;ID]).
Proof:
For alla Eg the function F belongs to F .[a]. So from Theorem 4.12 prOJ
it follows tbat for each a € g there exists a function Ga € S(R-2[a;ID])
such that Ga (x) = F(x) for all x E 11 a • ip(ID).
For all a,b E g we have that
because the functions Ga and Gb are continuous and their restrictions
tp 11 a • 11 b • ll)(ID) are equal (determined by F).
We define G on G. d[g] = u t 2 [a;ID] by J.n a€g
x E .e.2
[a; ID J , a € g •
Indeed, for alla Eg the restriction Gltz[a;ID] =Ga is analytic.
Since for each u,v E Gind[g] there exists a E g such that u,v E t 2 [a;ID],
the function À » G(u + ÀV) = Ga (u + ÀV) is entire. So the function G is
ray-analytic on Gind[g]. c
The fact tbat for each a € g the restriction GIR,2
[a;ID] is condnuous,
does not imply that Gis continuous on G. d[g], because the function G l.n
is not convex. Cf. Theorem I.2.3. However, under the additional condi-
tion that the conic Köthe set g is type II, we will prove that G is con
tinuous on Gind[g]. The proof relies heavily on the fact that the re
strictions GI Jl.z[a;ID] belong to S(Jl.2 [a;ID]). We give a proof in two
steps.
148 IV Spaces of analytic functions
S. 11. Theorem + Let g c w (:ID) be a conic Köthe set, let C denote the closed unit ball
in R.2 (ID) and let G be an extension of a function in Fproj[g] to Gind[g]
as in Theorem 5.10.
For each a,b E g and 8 > 0 there exists ÀO > O, such that
Proof:
Let a,b E g and let E > O. Since g is a conic Köthe set, there exists
c E g such that c ~ a,b. We will prove the strenger statement: There
exists ÀO > O, such tbat
vx,yEC : jG(c·x + À0c•y) - G(c·x>I < 8.
2 Put y = 118c Gii F' For all À > 0 we have
$ yjexp(x+ Ày,x+Ày) 2 - exp(X+ Ày,x) 2 +
- exp(x,x + Ày)2
+ exp(x,x>2i $
$ yjexp(x+ Ày,x)2
j • jexp(x+ Ày,Ày)2
- 1 j +
+ yjexp(x,x>2i • jexp(X,Ày) 2 - 1 j $
1+À À+À2 .À $ ye (e - 1) + ye (ë - 1) •
So, it is clear that we can choose ÀO so small that
jG(a•x·+ À0b•y) ~ G(a·x>I < E for all x,y E C
Clearly ÀO depends on Il Se G Il F'
s. 12. Theorem + Let g c w (ID) be a type II conic Köthe set and let G be the extension
of a function in F .[g] to G. d[g] as in Theorem 5.10. pro3 1n
Then the function G is analytic on G. d[g]. 1n
0
IV Spaces of analytic functions
Proof:
Let u E Gind[g] and let E > O, We prove that Gis continuous at u. We
assume that g ~ {am 1 m E lN} with an ;;.; an+t'
Let C denote the closed unit ball in .11,2 (ID). Since there exists b Eg -1
such that b • u E C, by Lemma 5.11 there exists À1
> 0 such that ·
VxEC : IG<u + >...1 a 1 • x) - G(u) 1 '< c./2 •
Inductively, we define a sequence (Àn)nElN in IR+:
149
Assume that we have found numbers À1, ••• ,Àm > 0 such that for ;;,; k;;,; m
(5. 1)
whe!te
v v : IG(u+ y+ À. ak• x) - Q(u+ y) 1 '< E/2k , xEC yEBk-l "k
k-1
~-1 = .l J=l
À. a .• C J J
k > 1 and {O} •
Put B = l~ 1 À.a.•C. Since gis a conic Köthe set, there exists C Eg, m J"' J J
such that u + Bm c: c • C. By Theorem 5. 11 there exists Àm+l > 0 sucb that
"xEC VyEB : iG(u+y+Àm+lam+1·x) - G(u+y)i < c./2m+1 • m
So we have found a sequence (Àn)nElN which satisfies (5.1) for all k.
Put V =co( U À a • C). Let v EV. Then there exist .\', E 1N, i3 > O, nElN n n .\', .\', m
x E C, 1 ;;,; m ;;,; R., such that \X. 1
B = 1 and v = \"' 1
i3 À a • x • m Lui= m L.m= m m m m
A simple application of the triangle inequality yields
IG<u+v) - GCu>I ~
R,
IG( u + ~ ) ( n-1 ;;,; l i3m Àa•x -Gu+l: B À a • x )1 n=1 m=1 m m m m=l m m m m
t ;;,; l c./2n :i e •
n=1
The set V is a convex, balanced subset of G. d[g], such that in
"vEV : IG(u+ v) - G(u) 1 < e •
For each n E lN the set V n i 2 [an;ID] is a neighbourhood of 0 in
;;,;
150 IV Spaces of analytic functions
t 2 [an;ID], so the set V is a neighbourhood of ~ in Gind[g]. Hence the
function G is continuous at u.
So fora type II conic KÖthe set g, the space F .[g] can be identified proJ
with a subspace of A(Gind[g]).
The space F . [g] bas the same structure as the projective limits prOJ
introduced in Section 2. Theorem 4.1 states that in general the space
Seq[F . [a]] is a proper subspace of t+2
[up[a];lM(ID)]. Nevertheless, prOJ
Seq[F . [g)] and G . [up[g)] are the same. This statement is similar proJ proJ
to the corresponding statement for Find[g) in Theorem 5.6.
s. 13. Theorem + Let g c w (ID) be a conic Köthe set.
Then the operator Seq maps F .[g) homeomorphically onto G .[up[g]]. proJ proJ
Proof:
Let F E F . [g]. For all a E g we have 9 F E F and hence Seq [9a F] = proJ a
= up[a] ~ Seq[F] E t2
(lM(ID)). So Seq[F] EG . [up[g]].
a
prOJ -l Conversely, let f EG .[up[g]]. For each a Eg put Ga= Seq [up[a]•f].
proJ Then Ga is an element of F and certainly analytic on \l)(ID). The restric-
tions of the functions 9a-1 Ga and 9b-1 Gb to 11a • 11b• \P(ID) are equal,
so the following definition makes sense:
The function F on \l)(ID) is defined by F(X) = Ga(a-1 •x), where a Eg with
ID[x] c ID[a]' x E \l)(ID).
It is clear that F is ray-analytic on \P(ID) and hence analytic on \l)(ID)
and Seq[F] = f. Since 9 F = G E F, we get F E F .[g]. Since for all a a proJ
F E F .[g) and a Eg proJ .
118a FllF·= llup[a] •_Seq[F]llR, (M(ID)) , 2
the operator Seq maps F .[g] homeomorphically onto G .[up[g]]. a proJ proJ
In the final part of this section we combine the inductive limit and the
projective limit viewpoint.
The Theorems 5.6 and 5.13 state that the spaces G. d[up[g]] and in
G .[up[g]), both contained in w(:M(ID)), are representations of the proJ
IV Spaces of analytic functions 151
spaces F. d[g] and F .[g], respectively, induced by the operator Seq. Ln proJ . These representation theorems have many topological consequences. Before
we summarize them, we introduce a pairing between the compound spaces
F. d [g] and F . [g ]. Ln proJ
5. 111. Definition + Let g c w (ID) denote a conic Köthe set and let (IDq)qElN be an exhaus-·
tion of ID.
The pairing <•,•>F between F. d[g] and F .[g] is defined by Ln proJ
<F ,G>f = lim '!T-q qElN
We note that qi(ID) and ~q are homeomorph. Cf. Corollary II.3.10. q
We remark that <F,G>F" <Seq[F],Seq[G]>, with <•,•>the pairing between
G. d{up[g]] and G .[up[g]] defined in (2.1). Ln proJ
We summarize the topological consequences of the representation theo
rems. We impose increasingly stronger conditions on the. set g.
(5.2) + Let g c w (ID) denote a conic Köthe set,
a. The space F. d[g] is barre led and bornological. Ln
b. The space Fproj [g] is complete.
(5.3) Let g denote a generating cone.
a. The inductive limit topology of F. d[g] equals the topology generated Ln by the seminorms ~, b. E (up [g ])# , defined by
'1, (F) " 1 b • Seq (F] 12 , F € F ind [g]
b. The space F. d[g] is nuclear iff Lll
152 IV Spaces of analytic functions
c. We regard the elements of F. d[g] as analytic functions on G .[g]. in proJ Let F E f. d[gJ and let B denote a bounded set in G .[g]. Let
ln # prOJ a E g such that 0 _ 1 F E F and let b € g such that B c b • C, where
The inductive limit topology of Find[g] is finer than the compact
open topology of Find[g]. Lemma I.2.3 implies that the seminorms
F 1+ sup IF(z) 1 , F € Find[g) , ze:B
with B a compact set, are continuous with r~spect to the inductive
limit topo logy.
d. The space F1.nd[g] is a representation of the strong dual of F .[g]. prOJ
e. The space F .[g] is semi-Montel iff prOJ
(5.4) Let g denote a #-symmetrie generating cone.
We regard the elements of F .[g) as ray-analytic functions on G1.nd[g]. prOJ
Let F € F .[g] and let B denote a bounded set in G1.nd[g]. Let a € g proJ
such that B c a • C. Then
So, the pré>jective iimit topology of Fproj[g] is finer than the compact
open topology of F .[g]. · proJ .
(5.5) Let g denote a generating cone such that up[g] is #-symmetrie.
A sufficient condition for this #-symmetry is the condition that g is
type II. Necessary is the condition that g is #-symmetrie.
a. The space Find[g] is complete and reflexive.
b. The space Fproj[g] is barreled, bornological and reflexive.
IV Spaces of analytic functions 153
c. The space F .[g] is a representation of the strong dual of f1.nd[g].
proJ
d. If VaEg VbEg#: a•b € c0 (ID), then both spaces Find[g] and Fproj[g]
are Montel.
e. If vaEg vbEg# are nuclear.
a•b € Jl1(ID), then both spaces F. d[g] and F .[g] in proJ
Appendix
As an illustration of the theory developed in this chapter we present
representations of modified Heisenberg. groups in compound spaces.
The classical (2n+1)-dimensional Heisenberg group .l1i is the Lie group
IR.n x IR.n x IR.[mod 27r] with group operation
The classical unitary representation UH of ~H in L2(IR.n) is given by
(A.2) iy+i(X,W)z
[UH(v,w,y)F](x) == e F(v+ x) •
The modified Heisenberg groups are groups with a group operation and a
representation, which are very similar to the group operation • and the
representation UH of the classica! (2n+1)-dimensional Heisenberg group
~H.
Consider the following sketch.
Let V and W be two vector spaces which are in duality with respect to
the sesquilinear form<•,•>. The group H(V,W) is the set V x W x t witt
the group operation
Cf. (A.1)
For each (v,w,y) € H(V,W) and each complex valued function F on V we
define the complex valued function U(v,w,y)F on V by
(A.4) [U(v,w,y)F](x) == ey+<x,w> F(x+ v) •
154 IV Spaces of analytic functions
Cf. (A.2). We have
So, if F0 is a space of functions on V, such that U(v,w,y) maps F0 into
F0, then U is a representation of H(V,W) in F0•
Consider the following operatörs:
(A.5) The translations Tv' v E V, with
[Tv F](x) • F(v + x) •
(A.6) The shifts Rw• w E W, with
CRw F]{X) = e<x,w> F(X) .
(A.7) The scalar multiplications Ey• y E C, with
[Ey F](x) = eY F(x) •
Then we have U(v,w,Y) = Eyo Rw 0 Tv.
Starting with a fixed conic Köthe set g c w+(:ID), we consider the
following two cases:
(A.8) CASE I:
(A. 9) CASE II :
F0 = F . [g]. proJ
We wil! prove that the translations TV and the shifts Rw are homeo
morphisms on F. d[g] and F .[g] for suitable V and w. The partial l.n prOJ
ordering of reproducing kemels is an essential tool in our proofs.
A.1. Lemma
Let u € Jl,2(ID). The complex valued function N(u) on qi(ID) x qi(ID) is
defined by N(u)(x,y) = exp[(x,u) 2 + (u,y) 2], x,y E qi(ID).
IV Spaces of analytic functions 155
Then N( ) is a function of positive type and for all À > 0 there exists u À11
y > 0 such that N(u) ~ yK
Proof:
Let À > 0. For x,y E (j)(ID) we have
R k (II 1 4) . l" h N < KÀll "th (1 !2/À2) emar • • i.mp i.es t at (u) ~ y wi. y = exp u 2 •
A. 2. Corollary
Let u E i 2(ID) and let À> 1.
a. The function (x,y) 1+ exp(u+x,u+y) 2 belongs to PT((j)(ID)).
b. For each À > 1 there exists y > 0 such that
Proof:
a. Let ,Q, E 1N and let aj E 4:, Xj E (j)(ID), 1 ~ j ~ ,Q,. Then
,Q,
I akaj exp(U+Xk,U+XJ.)2 k,j=t
0 .
b. By LeDmla A. 1 there exis ts 8 > 0 such that N ( U) ~ 8 KIA-T 11 •
LeDmla II.t.20 implies that
lu1 2
with y = 8e 2
a
a
156 IV Spaces of analytic functions
CASE I
We consider the group G .. d = H(G .[g],G. d[g]) and its represen-proJ,in pro) in tation (v,w,y) 1+ Ey~ Tv, further denoted by U, in the compound space
F. d[g]. Theorem 5.5 implies that we can extend the elements F of in F. d[g] to analytic functions on G .[g] in precisely one way. In the in proJ sequel we regàrd the functions F in F. d[g] both as analytic functions in on G • [g] and as analytic functions on (j)(lD).
prOJ
We start with a study of the translations T, v EG .[g], and the V prOJ
shifts R , WEG. d[g], on F. d[g]. For each F E F. d[g] the functions "'W in in in Tv F and ~ F are well defined on (j)(lD), Our first step is to prove that
the operators TV and ~ map elementary spaces continuously into elemen
tary spaces.
A. 3. Theorem
Let a,b E w+ (ID), v E .e.;ca,ID]' w E R,2 [b;ID].
a. For all À> 1, the translation Tv maps Find[a] continuously into
F. d[Àa]. ln
b. The shift ~ maps Find[a] continuously into Find[a+b],
Proof:
a. Let F E F. d[a]. Theorem 4.6 implies that we can regard F as a func-in +
tion on i;ca;ID] such that F(X) = F(11a • x) for all x E t 2 [a;m]. ·
Using Lemma II.1. 7 we prove that Tv F belongs to Find[Àa]. Let !/, E JN,
a. Et, x. E (j)(ID), 1 ~ j ~R.. Since 0 -1 F can be extended toa J J a
unique element of S(!l2(ID)), we get
R. 2 R, 1 I et. [Tv F](x.) 1 • 1 I a. F(V + x.) 12
= j=1 J J j•1 J J
Il,
lj!1
aj[ea_1 FHa•v+a•xj>l2
:;;
!/,
~ 110 _1 Fll~ l exp(a•v+ a•xk' a•v+ a•xj)Z • a k,j==1
IV Spaces of analytic functions 157
Let À > t. Sincec a • v E i2
(ID), by Corollary A.2 there exists y > O,
depending on a • v, such that
R, 2 2 R, 2 1 l: et. [T F](x.) 1 ;; y • 110 _1 FN l: exp[À (a•xk.a•xJ.) 2 ] • j=t J v J a f k,j=l
Hence Tv F E find [Àa] and
b. Let F E Find[a]. For all x E q>(lp) we have ['\ F](x) = exp(<x,w>) •F(x).
The function x i+ exp(<x,w>), x E q>(ID), belongs to f. d[b]. in Theorem 5.2 yields that '\ F belongs to find[a+b] and
111\ Fllf. [a+b] ;?; exp(llb-1
• wl~) • llFllf. [a] • ind ind
a
A. 4. Corollary
Let g be a conic Köthe set, V EG .[g] and WEG. d[g]. prOJ . l.n The translation Tv and the shift Rw are homeomorphisms from find[g] onto
F. d[g]. in
Proof:
Since g is a cone, the translation Tv. maps f. d[g] into f. d[g] and for . l.n in . each a Eg the restriction TvlF· [a] is. continuous. Lemma I.2.3 yields ind _
1 that Tv is continuous on find[g]. Further, (Tv) " T_v• So the trans-
lation Tv is a homeomorphism from F. d[g] onto f. d[g]. in in The proof for the shift '\ runs similarly. a
A.s. Corollary
Let g be a conic Köthe set.
Then U represents the group G . . d as a group of continuous linear proJ,l.n operators on Find[g].
In the appendix to Chapter II we have introduced for each u E H the
annihilation operator a(u) and the creation operator y(u) in S(H). For
all suitable F E S(H) and x E H we have
158 IV Spaces of analytic functions
[a(u)F](x) = lim (F(x +Au) - F(u))/;\ , ;\-!{)
[y(u)F](x) = (x,u)H F(x)
Moreover, we have seen in Appendix II.A that exp(a(u)) = T and u
exp(y(u)) = R • u
Tbis bas been our motivation to introduce for v € G .[g] and proJ w E Gind[g] the operators a(v) and y(W) on Find[g] by
[a(v)FJ(x) = lim (F(x + Av) - F(x))/À , À--+Ü
[y(W)F](x) = <X,W> F(x) •
It.can be proved that a(v) and y(w) are continuous operators from
Find[g] into Find[g]. The proof runs the same as in the cases of Tv and
~· We remark that a(V) is the inf initesimal generator of the group
{TÀV 1 À € t} and y(W) is the infinitesimal generator of the group
{RÀW 1 À € 4:}.
CASE II
We consider the group G. d . = H(G. d[g],G .[g]) and its represen-in ,proJ in proJ tation (v,w,y) 1+ E R Tv• denoted by U, in F . [g]. Theorem 5.10 states
. y·• ~~ that each F € F .[g] bas a unique ray-analytic extension G to G. d[g] proJ in such that for alla Eg the restriction G!t
2[a;ID] is analytic. In the
sequel we regard the functions in F .[g] both as ray-analytic funcproJ tions on G . [g] and as analytic functions on q>(ID). prOJ
We consider the translaÜons Tv• V E Gind[g], and the shifts ~· w € G . [g]. For each F € F • [g] the functions Tv F and R F are
proJ proJ ·• well defined on q>(ID).
A. 6. Theorem
Let g be a conic Köthe set, V € G. d[g] and W € G .[g]. ln proJ The translation Tv and the shift~ are homeomotphisms from Fproj[g]
onto F . [g]. prOJ
IV Spaces of analytic functions 159
Proof:
Let F € F .[g]. By definition the.function T F belongs to F .[g] proJ . v . proJ
iff for all a E g the function 0 [T F] belongs :to F. a v _1
So, let a E g. There exists b E g such that b ~ a and b • v E i 2 (ID).
We use Lemma II.1.7 in order to prove that 0a[Tv F] E F. Let À> 1, R, € JN and let aj E 4:, xj E (f)(ID), 1 :il j :il R,. Put
R, 2 cr = 1 t aJ. ea [Tv F](xJ.) 1 •
j=1
Then
R, 2 jl, . 2 cr = 1 t aJ. [Tv F](a•xJ.) 1 = 1 t a. F(v + a•x.51 =
j=1 j=1 J J
R, . -1 -1 -1 -1 . 2 l Z: aJ.[0Àb F](À b •v +À b •a•xj>I :il j=1
-1 By Corollary A. 2 there exists y > 0, depending on À b • v, such that
b-1·a Because K :il K, we find
Hence 0a [Tv F] E F and
We conclude that the function Tv F belongs to fproj[gl and that the
translation TV maps F · .[g] continuously into F .[g]. Since _1
prOJ proJ . (Tv) = T-v' the translation Tv is a hom.eomorphism.
f60 IV Spaces of analytic functions
Let F E Fproj[g]. The function x >+ exp(<x,w>), x € ~(1D), belongs to
Fproj [g]. Theorem 5.9 implies that !\, F E Fproj [g] and
The remaining part of the proof runs ·similarly as in the previous case. c
A. 7. Corollary
Let g be a conic Köthe set.
Then U represents the group G. d . as a group of continuous linear i.n ,proJ operators on F . [g]. prOJ
As before, we define the operators a(v), v EG.- d[g], and y(w), in
w E Gproj[g] by
[a(V)F](x) lim (F(x + ÀV) - F(x))/l. , À>O
[y(w)F](x) = <x,w> F(x) ,
x E G • d [g] , F E F . [g] • l.n prOJ
They are continuous operators from F .[g] into F .[g] and they are proJ proJ the infinitesimal generators of the groups {î ÀV 1 À E C} and
{RÀW 1 À E C}, respectively.
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absorbing analytic function analyticity space annihilation operator asymptotically equivalent
balanced Bargmann space
of finite order of infinite order
barrel barre led bornological
canonical injection Cauchy net collection of seminorms
This thesis is a treatment on spaces of analytic functions on sequence
spaces. Both the sequence spaces and the analytic function spaces belong
to a class of locally convex spaces, which are inductive limits of
Hilbert spaces, projective limits of semi-inner product spaces or both
at once.
By means of these analytic function spaces the concept of symmetr-ia
Fock 8pace for Hilbert spaces is generalized to sequence spaces in our
class. In this way, natural Fock space constructions can be carried out
for e.g. tèst spaces and distribution spaces.
Our treatise is an amalgamation of ideas from the theory of analytic
f.unctions on locally convex spaces, from the theory of the reproducing
kemel and from a recent study of inductive/projective limits of Hilbert
spaces.
As an application we construct representations of infinite dimensional
Heisenberg groups.
SAMENVATTING
Dit proefschrift gaat over ruimten van analytische functies op rijtjes
ruimten. Zowel de beschouwde rijtjesruimten als de ruimten van analy
tische functies behoren tot een klasse van lokaal convexe ruimten, die
inductieve limieten van Hilbertruimten, projectieve limieten van semi
inproduktruimten of beide tegelijk zijn.
Uitgaande van deze ruimten van analytische functies wordt het concept
symmetrische Fockruimte voor Rilbertruimten veralgemeend voor rijtjes
ruimten in onze klasse. Op basis van dit algemene resultaat kunnen dan
op een natuurlijke manier Fockruimte-constructies worden uitgevoerd
voor bijvoorbeeld testruimten en distributieruimten.
Onze beschouwingen zijn gebase.erd op ideeën uit de theorie van analy
tische functies op lokaal convexe ruimten, uit de theorie van reprodu
cerende kernen en uit een recente studie over inductieve/projectieve
limieten van Hilbertruimten.
Bij wij ze van toepassing worden representaties van oneindig dimensio
nale Heisenberggroepen geconstrueerd.
22 juni 1951
juni 1970
juli 1982
augustus 1983 tot
september .1987
vanaf september 1987
CURRICULUM VITAE
geboren te Lith
eindexamen gymnasium-S Bisschoppelijk College
te Roermond
doctoraal examen wiskundig ingenieur T.H.E.,
met lof
wetenschappelijk assistent T.H.E., vakgroep
analyse
medewerker bij de groep Basisonderwijs,
faculteit Wiskunde en Informatica, T.U.E.
STELLINGEN behorende bij net proefschrift
SPACES OF ANAL YTIC FUNCTIONS ON lNDUCTIVEIPROJECTIVE LIMITSOF HILBERT SPACES
door F.J.L. Manens
L De stelling van Kleinecke-Shirokov. 1 K 1, heeft de volgende asymptotische uitbreiding~
Zij A een compacte operator op een Hilbcrtruimte H_
Voor ledere e > 0 bcMaat er een ~ > 0 zodar1ig dat voor alle B " B(H) met 11 [A , fA , 8 :1 ,111 kleiner <ian S /11811 de spectrale straal van [A. 8 J kleiner dan t is.
2. Beschouw het Gelfandtripd
SL,(S,l.•~ '=--> L,(So) C...., T~,(S,).o::O
van respectievelijk de analytische futiCties, de kwadratisch integreerbare functies en de hyperfuncties op de eenheidssfccr S2 in !R 3 _ VgL [ G 1. Zij[, de oplossing van het Dirichletproblcem op de eenheidsbol met randvoorwaarde f" L,(s,,_, ), Zij I' de projectie in IR 3 met f'(..: 1 , x2 , x3)" (x 1 • D. DJ-
De operator /.,p met ll.r f 1 (i;)= J.(P ~) beeldt SL,(S,),d)l, continu in SL,(.>,).•~ af, maar i.~
niet uit te breiden tot een continue operator van Tus,) .•• ~ in 'f'L,(.ç,),t.!'..
3. Zij !>2(Sq_1) de Bilhertruimte van kwadratisch integree•·bare functies op de eenheidssfeer s •. 1 in JR• _Zij voorts SL(IR, q) de groep van reële q x 'f matrices met determinant 1. De afbeelding ot>, gegeven door het voorschrift
is een unitaire representatie vatl SI.( IR, q) in L2(s._1).
Deze represetltatie is in-educibel op zowel de even als de oneven dervlruimtc in Lz(Sq-1)-
De operatoren tl>(A) zijn uit te breiden tot continue bijccties op de ruimte van hyperfuncties op s,_,'
4. Laat A een positieve :>:elfgeadjungeerde operator op 12 zijn met de eigenschap dat e-•A een Hilben-Schmidt operator is voor alle 1 > 0 en C de positieve zclfgcadjungccrdc operator op de Bargmannruimte B~ met ,__,c F"" F o •-'", FE B~, 1 > 0. De ruimte s~_.c bestaat uit precies die analytische functies F van T1,.t. in (f met de
eigenschap
3,.0 3">Q 'i_.,, : I F(z) I :;; ~ exp (~ I ,-•A z I i).
De duale ruimte T&..c bestaat uit precies die analytische functies F van s,.,, in I[
met de eigenschap
0',,0 ~,.,.o 0',,1, : I F(e-IA z) I :;; ~ exp <t I z I~).
Zie[M].
5. Beschouw de HUbertruimte Hz(B") van kwadratisch integreerbare harmonische functies op s,. de eenheidsbal in m•. De reproducerende kern K van H21Jl.l is
(I-I x Ij I y I~/ K(l,y)=Q (l+lx I~ ly lf:-(;:-y}z)f;+i-- (I+ I x I i ly 1~-(:<,yh)i-•.
De uitdrukking voor K, die vermeld wordt in [L], is niet correct.
6. Zij 1,, v > o, een semigroep van quasinilpotente operatoren op een Banachruimte x. Dan is voor iedere x eX, .t;<O, de verzameling {J,x I v > 0) onafhankelijk.
7. Zij S(IR") de Schwartzruirnte van snelafnemende functies. Zij A een dient gedellnieenle opel'ator in L 1(R") met de eigenschap datA"(S(IR")l c:S(IR"). Dan geldt
(a) A • is een continue afbeelding van de testruimte SC IR") in zichzelf_
(b) A i~ uit te breiden tot een continu~ afbeelding van de ruimte S'(IR") van getemperde distributies in zichzelf.
Vergelijk [E)_
8. Zij 1 een geheel analyti se he functie met positieve Taylorcoëfficiëntcn_ Dan is (-< , y) ,____, f((x , y )z!, x , y e 1 "' een reproducerende I<ern op 1, _ In de bijbehorende functionele Hilbcrtruimte liggen de harmonische functies van eindig veel variabelen overal dicht. Vergelijk Stelling ll.3.28 van dit proefschrift.
9. ledere vol nudeaire Kötheruimte is van de vorm C,""[l:l- Hierin is g een #symmetrische Köthevcrzamding.
Literatuur
IEl Eijndhoven, SJ.L. van, A theory of generalizcd functions based on one paran•eter gro~,~ps of unboundcd sclf-adjoint operators, TH-Report 81-WSK-03, Eindhoven University of Technology, Eindhoven, !981.
[G] Graaf,]_ de, Two spaces of generalizcd functions ba.sed on harmonie polynomials, in C. Brezinski, etc. (ed.), Polynömes orthogonaul\ et Applications, f>roc_ Bar-lc-Duc 1984, !eet. note~ in Math. ll7l. Springer-Verlag, 1984_
]KJ Kleinecke, D.C., On oper~tor commutators, Proc. A.M.S. 8 (1957), blz. 482-486.
[L] Ligocka, E., On the reproducing kemel for hat-monic functions and !he space of Bloch harmonie functions oo lhe unit ball in IR", Stud. Math., 87 (1987), blz. 23-32.
[M] Mancns, f'.J.l.-., 1-'unction spaces of harmonie and analytic functions in infinitely many variables, 8UT report 87-WSK-02, Eindhoven Univcrsity of T(;chnology, Eindhoven, 1987.