Connected components in the space of composition operators onH∞ functions of many variables

Integr. equ. oper. theory 45 (2003) 1-14 0378-620X/03/010001-14 �9 Birkh~user Verlag, Basel, 2003

I Integral Equations and Operator Theory

C O N N E C T E D C O M P O N E N T S I N T H E S P A C E O F C O M P O S I T I O N O P E R A T O R S O N H ~ F U N C T I O N S O F M A N Y V A R I A B L E S

RICHARD ARON, PABLO GALINDOtand MIKAEL LINDSTROM t

Let E be a complex Banach space with open unit ball BE. The structure of the space of composition operators on the Banach algebra H~176 of bounded analytic functions on BE with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly inside BE form a path connected component. When E is a Hilbert space or a C0(X)-space , the path connected components are shown to be the open balls of radius 2.

Let E denote a complex Banach space with open unit ball BE and let r : BE --~ BE be an analytic mapping, where F is also a complex Banach space. We will consider composition operators Cr defined by Cr = f o r acting from the uniform algebra H~C(BF) of all bounded analytic functions on BF into HO~ Our object of s tudy is C(H~176 HC~ the space of all composition operators endowed with the operator norm topology.

Motivated by earlier research of Shapiro and Sundberg [19] on the space of composition operators on the Hardy space H 2, MacCluer, Ohno and Zhao [15] characterize the connected components and the isolated points in the space C(H ~176 H~). Their work was ex- tended in [10] to the space of all endomorphisms of H ~176 In the H ~ setting, the main result is that the (path) connected components are the open balls of radius 2. A key tool to obtain these results is the use of the so-called pseudohyperbolic distance in the ball BE, which is just the Poincar5 pseudodistance in the case of the unit disc. There is no known (to the authors) formula for the pseudohyperbolic metric in general Banach spaces. Nevertheless, for some special Banach spaces such as Hilbert space or a Co(X) space, an explicit formula is known. Therefore, in these cases, we can prove that the open balls of radius 2 are the path connected components in C(H~176 H~176 and also that the composition operators arising from biholomorphic mappings are isolated points, a result which holds for a wide class of Banach spaces. In [15] it is shown that in the H ~176 case, the set of composition operators which differ

*The research of this author was supported by grant number SAB1999-0214 from the Ministcrio de Educacidn, Cultura y Deporte during his stay at the Universidad de Valencia.

*The research of this author was partially supported DGES(Spaln) pr. 96- 0758. *The research of this author was partially supported by Magnus Ehrnrooths stiftelse.

2 Aron, Galindo, Lindstr6m

from a given one by a compact operator belongs to an open ball of radius 2. In this paper we extend this result by proving, in the JB*-tr iple setting, that two composition operators whose difference is a completely continuous operator also lie at norm distance less than 2. Despite our lack of an explicit general formula for the pseudohyperbolic distance, we can show that , no mat te r which infinite dimensional Banach space E we deal with, the set of composition operators arising from mappings r with range strictly inside the unit ball is its own (path) connected component, and hence the set of compact composition operators do not form a connected component.

P r e l i m i n a r i e s . The reader is referred to [6] and [17] for background information on analytic functions on an infinite dimensional Banach space. The algebra Hc~ is a Banach algebra with the natural norm IIfll -- supzeB~ If(x)l . This algebra, which is a natural generalization of thc classical algebra H ~176 of analytic functions on the complex open disk A, has been studied in [2], [4], [5], [10], and [3]. Let M(H~176 denote the maximal ideal space of H~176 tha t is, the space of all complex homomorphisms on H~176 As usual, the pseudohyperbolic distance p(m, n) for m, n E M(H~176 is defined by

p(m, ~) = sup(I](~)l : ] e H~176 Itfll < 1, ] (m) = 0),

where f is the Gelfand transform of f . For x E Bz, dx denotes the point evaluation at the point x, i.e. 5~(f) = f(x) for all f e H~~ Clearly 5~ �9 M(H~176

For any complex Banach space E denote by s the Banach algebra of all bounded linear operators from E to E, and let Aut(BE) denote all biholomorphic maps from BE into BE. For a locally compact Hausdorff space X, we let Co (X) denote the C*-algebra of complex valued continuous functions on X which vanish at infinity, endowed with the sup-norm.

A JB*-triple ( see [13] or [7]) is a complex Banach space E with a continuous triple product {.,., .} : E • E • E --4 E which is bilinear and symmetr ic in the outer variables, and conjugate linear in the middle variable, and which satisfies the following two conditions:

(i) the operator z ~+ {x, x, z}, denoted x o x, is hermitian with non-negative spec- t rum for all x �9 E and I[xoxl[ = IIx[[2;

(ii) the product satisfies the "triple identity"

{a, b, {x, y, z}} = {{a, b, x) , y, z} - {x, {b, ~, y}, z} + {x, v, (a, ~, z } }

In this definition I l x o x l l = I1~112 c a n be replaced by I]{x,x,x}l] = IIxl[ 3 ([13] p. 523) and furthermore [[{x,y,z}l I ~_ [Ixll [lyI[ IIzII for all x ,y , z �9 E ([12] p. 278). The complex plane is the simplest example of a JB*-triple with the triple product {x, y, z} = x~yz. Any C*-algebra is a JB*-triple with the triple product {x, y, z} = �89 + zy*x).

The layout of this article is as follows. In w we prove general results concerning components in the set C(H~176 H~176 of composition operators. We also discuss components which are singletons (i.e. isolated points). In w we specialize to the case in which F is a JB*-triple, obtaining in particular a characterization of the connected components of C(H~176 H~176 for F a Hilbert space or a commutat ive C* algebra.

Aron, Galindo, Lindstr6m 3

w G e n e r a l resul t s . In this section we show that the set of composition operators arising from mappings with range strictly inside the unit ball is a (path) connected component in C(H~176 H~176 Later we provide conditions for the existence of isolated points in

Let H~176 BF) denote the space of all analytic functions from B~ into By.

L e m m a 1. The map C: H~176 BF) ~ C(H~176 H~176 which assigns to every 0 the composition operator Cr is continuous at every r 6 H~176 BF) such that r C rBv for some O < r < l.

Proof. Let r 6 H~176 BF). Then

l I C e - Cr = sup I I h o r h o r I = sup sup [h(r - h(r < []h[[<l ][h[J<l ZeBE

2 l i e ( x ) - r < 2 l [ r r sup

�9 1-11r - 1 - r by using inequality (2.1) in [4]. �9

In general the mapping C in the above lemma is not continuous at flmctions r which do not ,nap BE strictly inside By. Indeed, i r E = C = F, and Cr(x) = rx, 0 < r < 1, we have that Cr --~ id as r -+ 1- in H~176 BE). However, Cr does not converge to Id in C(H ~176 H ~176 since every CCr is compact. A more general result is indicated in Proposition 5, below.

R e m a r k 2. As [[Cr - C r = sup,eu t l lS<~) - ~r it follows from [14], 1.8 Satz that

alIC - Cell = sup p(~r162

4 +IICr - c r =

where p denotes the pseudohyperbolic distance in M(HC~

P r o p o s i t i o n 3. In C(H~176 H~176 the set of composition operators which are compact (respectively, which arise from mappings r with range strictly inside BF) is path connected. Proof. First, by [3], Prop. 3, Cr is compact if and only if the range o f r is a relatively compact set strictly inside BF. If r and ~b 6 H~176 BF) have their ranges strictly inside BF, then any convex combination Ar + (1 - A)r has its range strictly inside BF as well. Moreover if r and r 6 H~176 BF) have relatively compact ranges, the range of Ar + (1 - A)r is relatively compact. Since the mapping A E [0, 1] --4 Ar + (1 - A)r E H~176 BF) is trivially continuous, the above lemma applies to show that the mapping A 6 [0, 1] --+ C~r162 6 C(H~176176176 is continuous and maps the interval [0, 1] into the set of compact composition operators. So this is a path connected set.

Replacing relative compactness by the condition of being strictly inside BF, we obtain the corresponding result. |

L e m m a 4. Ire and r : BE -+ A are such that [[Cr - Cr < 2 and one of the associated composition operators is compact, then the other is also compact.


Proof. Say Cr is compact. Pick c such tha t Ilc - c , II < c < 2. Then for every x E BE, we have 115r -fir < c. By Remark 2 there is d < 1 such tha t for the pseudohyperbolic distance p in M(HC~

r - r

Assume tha t r is not relatively compact, so tha t there is a sequence {x~} C Bz such that {[r converges to 1. There is no loss of generality in assuming tha t limn ~b(xn) = exp(i0). Since Cr is compact, the set r is a relatively compact set in A. Hence we can extract a subsequence of {xn} such that {r converges to some a e A. Then, letting k --+ cx~, we have

r r --~ I a - exp(iO)

which is impossible. |

P r o p o s i t i o n 5. Suppose that r and r are analytic mappings from B~ to BF and that r C rBF for some O < r < 1. I f either a) K := Cr - C o is a completely continuous (or weakly compact) operator or b) I IC , - C~I I < 2, then r lies strictly inside BF as well.

Proof. Assume not, i.e., assume that there are xj E BE such tha t ][r -+ 1. Put zj = r c F and, as in the proof of Thin 10.5 in [2], choose Lj E F* with []Lj[[ < 1 and �9 j, a conformal map of the open unit disc onto the right half plane, to construct the function fj = (I~j o L i. If I[z[I _ r, then [[L~(z)[[ <_ r, and hence f j (rBv) is a relatively compact set in the right half plane. Moreover, as observed in [2], the series ~ f j uniformly converges in rBv, and hence ( ~ f j ) ( r B v ) is also a relatively compact set in the right half plane. Therefore the

maps rBv into a compact set in A. Further, lira d g(zj) = 1. Note that function g -- ~ l j + l

C~or is compact since g o r is a relatively compact set in the open unit disc. a) We consider the corresponding mappings Cgor K o Cg, and Coo ~, from H ~176 to

H~~ note tha t K o C 9 is completely continuous (resp. weakly compact) . Now, as Cr = K + Cr we have Cr o g) = K ( f o 9) + Cr o g). Hence C9or =

(I( o C~)(f) + Cgor which shows that C~or is a completely continuous (resp. weakly compact) operator. But then g o r would have relatively compact range in the open unit disc ([9], Prop. 2) and this contradicts the fact that 9 o r = g(zj) has an interpolating subsequence for H ~176

b) Now, I l C ~ o r - C9or = Ilcg o (cr - ce) l l _< I Ic r C~ll < 2, and thercfore, IIGor162 < 2. Hence we may apply Lemma 4 and conclude that, since C~o~ is a compact operator, Cgor is a compact operator as well. Thus we are led to the same contradiction as in a). |

We will show after Theorem 14 that assumption b) in the above proposition does not imply assumption a). We do not know whether in general assumption a) implies assumption b).


C o r o l l a r y 6. The set of composition operators arising from mappings r with range strictly inside BF coincides with its own path connected component.

Proof. As the open balls of radius 2 are clopen sets in C(H~176 H~176 ) ([10], Prop. 9), the components lie inside such balls. Then Proposition 5 b) completes the proof. �9

Whenever E is infinite dimensional the set of compact composition operators in C(H~176176176 is not its own connected component K:; for example if r = }, Corollary 6 shows that the non-compact operator Cr E K:.

As expected, knowledge about the structure of C(H~~ ~~ may be easily used to obtain information about the structure of C(H~176 H~176 For instance, for any functional # E E*, []#1[ = 1, and any non-zero element x E BE, we may consider the mapping F given by

r E BHoo ~ x . r o ~ E H~~ BF).

Actually, F is one to one since F(r = F(r implies x . r o #(e) = x . r o j~(c) for all e E BE. Hence r ---- r and thus r = r as # takes all values in A. Then the continuous mapping

C~ E C(H ~176 g ~176 ~-~ Cr(r E C(H~176 H~176

is also one to one, and therefore the connected sets and the non-isolated points in C(H ~176 H ~176 are taken to connected sets and non-isolated points in the space C(H~176 H~176 respectively.

Corollary 6 gives one of the connected components of C(H~176 H~176 al- though the set of composition operators arising from mappings r with range strictly inside BE is far from exhausting all them. This follows both from the next proposition and also from the fact that there are functions r E Bgo~ whose range does not lie strictly inside A for which the associated composition operator Cr is not isolated ([15], Cor. 9).

Our next goal is the search for isolated points in C(H~176 , H~176 ). For this, the following definition will be convenient. For an algebra "4 of functions defined on BF, a set D C BF will be said to be an , 4 - d e t e r m i n i n g set if every f E ,4 such that lim=_~j f ( x ) = 0 for all y E D vanishes on BF. This definition is motivated in part by the observation that in the case of the complex plane, any subset of the unit circle having positive measure is an H~176 set. Several other examples of A-de te rmin ing sets will be given after the following lemma.

Recall that A,,(BE) denotes the (closed) subalgebra of H~176 of uniformly continuous functions.

L e m m a 7. Suppose that D C BF is Au(Bp)-determining and that for some subset L of the unit circle of positive measure, AD C D for every A E L. Then D is an H~176 set.

Proof. Suppose that f E H~(BF) is such that limz_,,j f (x ) = 0 for all y E D but that f (v) ~ 0 for some v E Bp. Define the function g in BF by g(u) = f(au) for some fixed a, 1 > a > II~ll- Since f is uniformly continuous on balls of radius less than 1, g is uniformly

6 Aron, Galindo, LindstrSm

continuous on the closed ball. Also, since g(~) = f(v) # O, there is w �9 D such that g(w) ~ O. Then the function on A given by h(z) :-- f (zw) belongs to H ~176 and for every A �9 L, liln,_,A h(z) = lim,~_~x~ f (zw) = 0 since Aw �9 D. So h vanishes on L and hence is a null function. In particular, 0 = h(a) = f(aw) = 9(w), which is a contradiction. I

By the ma~ximum modulus principle, the unit sphere SF is A~(BF)-dctermining, and hence SF is also H~176 Also if F = gp, 1 < p < +(x~, the set D = {v �9 Stp: v is finitely non-zero} is H~176 For the polydisc Be- , the torus T ~ is an H~176 set, since it is a determining set for the polydisc algebra (see [18] for this and other examples of determining sets for the polydisc algebra). The same holds for the infinite dimensional torus in goo ([1], Thm. 6).

D e f i n i t i o n 8. A point y �9 Sr is said to be a w - s t r o n g p e a k p o i n t for ,4 i f there is h �9 A, [[h[[ = 1, such that the following holds: limx-~y h(x) = 1, and for every weak neighbourhood W of y, there is b > 0 such tha~ [h(x) - 11 > b fo r al1 x �9 BE \ W.

E x a m p l e s 9. a) If F is a strictly convex reflexive Banach space, every point y in the unit sphere is an H~ w-strong peak point: Indeed, as F is strictly convex, there is u � 9 F*, ] [u [ I= 1, which peaks at y ; t h a t i s , u(y) = 1 and i fu (x ) = 1 for s o m e x �9 BF, thcn x = y. If for some weak neighbourhood W of y, the condition failed, there would exist a sequence (x,) C BF \ W such that lira, u(x,) = 1. The reflexivity of F leads to a weak cluster point xo of (xn) for which U(Xo) = 1. Hence xo = y, and many of the (xn) would then belong to W.

b) Every strong peak point for A,(BF) is a w-strong peak point for H~~ The proof of Theorem 2 in [1] shows that every finitely non-zero vector v �9 S~,, 1 < p < +oc , is an A,(B~,) strong peak point, hence an H~176 w-strong peak point.

c) Similarly, one can see that every point in the torus T n C C" is an H~~ .) w- strong peak point, by checking that it is actually a strong peak point for the polydisc algebra. Actually these are the only w-strong peak points for H~176 : Indeed, if ( z l , . . . , z n ) is such a point and one of its coordinates has absolute value less than 1, say Iz, I < 1, we pick a sequence (a,~) c A n-1 converging to (Z l , . . . , z , -1 ) and consider the analytic functions hm(z) := h(am, z), z �9 A. By Montel's theorem there is g �9 H ~176 such that g is the pointwise limit of a subsequence, which we still denote (hm). Then since ]gl < 1 and g(zn) -= limm h(a,~, z,) = lim( . . . . . )-~( ......... ) h(am, z,) = 1, g must be the constant function 1. Since (zl . . . . . z ,) is a w-strong peak point, for r = 1 - [ z n I and W = B( ( z l , . . . , z~), r), there exists b > 0 so that ]h(x) - 11 > b for all x �9 Be , \ W. Now for any u �9 A with I u - z,] > r, the point (am, u) ~ W, so ]h(am, u) - 11 > b. This contradicts our hypothesis

that lim,~ h(arn, u) = limm hm(u) = g(u) = 1. d) There are no w-strong peak points for H~176 Indeed, we claim that if (z,) �9

S~o were a w-strong peak point, then for every k, sk = (zl, ..., zk) would be a w-strong peak point for H~176 By part c) above, it would follow that for every k, sk would belong to T k, contradicting the fact that (zn) �9 co. To prove the claim, suppose that we have


found a function h 6 H~176 satisfying Definition 8. Then the mapping g defined by g(ul , . . . , uk) := h(ul , . . . ,Uk, Zk+1, zk+2,...) would be a norm one element of H~176 ) and

lim g(ul , . . . , uk) = lim h(ul , . . . , uk, za+l, zk+2,...) = 1. (~a,...,u~)-~sk (ul,...,~k)~sk

Moreover if ( z l , . . . , xk) 6 Bck\B(sk, r), we would also have that ( x l , . . . , xk, Zk+l, Zk+2,...) r W where W = {(a,) 6 co : [ a i - z d < r, i = 1 , . . . , k } is a weak neighbourhood of (z~). There would have to exist a b > 0 such that b < [h - 1[ in B~ o \ W; hence ]g(zl , . . . ,xk) - 1[ = [h(xl , . . . ,Xk, Zk+l,Zk+2,...) - - 1[ > b. This would imply that sk is a w-strong peak point.

P r o p o s i t i o n 10. Let r : BE --+ Br be an analytic mapping, and let D C E be defined ,as

D := {y 6 SE : r has continuous extension r to y and r is an H~ w-strong peak point} .

I f D is an H~176 set, then Cr is an isolated point in the space of all composition operators C(H~176 H~176

Proof. Assume IICr - Cr < 2 for some r : BE --+ BF. By Remark 2 there is d < I such that sup~eB ~ p(5r 5r < d. We will show that for all y 6 D, the norm-to weak limit lim,-4,a r = r

So let (xn) c BE be a sequence which converges in norm to y. If (r does not converge weakly to r there is a weak neighbourhood W of r such that r ~ W for infinitely many n ' s . Since r is an H~176 w-strong peak point, there are b > 0 and h 6 H~176 with limw.~r h(w) = 1 such that [h(r - 1[ > b for infinitely many n's. Since holomorphic mappings are non expansive for the pseudohyperbolic distance, it follows that P(~h(r ~h(r < p(6r (fr < d for x E BF. By passing to subsequences, we may assume that {h(r is convergent to, say, c 6 /~. Also recall that lima h(r = 1. Whcn using the fact that P(hh(r ~h(r < d and taking limits, we have that p(5~, 51) _< d. Hence c = 1, which is a contradiction. Therefore, (r must converge weakly to %b(y), as wc wanted.

Now for every u 6 BF., the function u o (r - r belongs to H~176 and

lim[u o (r - r = l i~u ( r - limju(%b(x)) = O. x--~y"

Then the assumption of D being H~176 leads to u o (r - r = 0, and hence r162 m

P r o p o s i t i o n 11. Let F = Co(X). The identity is an isolated point in the space of all composition operators C( H~( BF), H~176 BF) ).

Proof. Assume [ICe - id[I < 2 for some r : BF -+ BF. Let x E BF and s E X. We aim to show that r = x(s).

8 Aron, Galindo, LindstrSm

For e > 0, one can find a compact neighbourhood V of s and a relatively compact open set L such that V C L C {t �9 X : Ix(t) - x ( s ) l < e}. Then by Katetov's theorem, there is he �9 Co(X) such that Xv <- he <_ XL. Let f : A ~ BF be given by f ( z ) -- z(1 - he) + zh~ and let e be the function e : y �9 BE --+ y(s) �9 A. Both are well-defined analytic mappings. If g = e o r o f , we have that for the corresponding composition operators, C~ = C: o Cr o C~. On the other hand, C I o Id o Ce = Id since for any u �9 H ~176 [C I o Id o Ce] (u)(z) =

( u o e o f ) ( z ) - - ( u o e ) ( f ( z ) ) = u ( f ( z ) ( s ) ) = u ( x ( s ) ( 1 - h e ( s ) ) + z h E ( s ) ) = u ( z ) .

Now as I]Cf o Cr o Ce - C I o Id o CeI ] < ][C o - Idll < 2, it follows from Proposition l0 (cf. [15], Cor. 8) applied to F = C = E and the identity mapping that g = id; tha t is z = (e o r o f ) (z ) = [(r o f)(z)](s) = r - he) + zh~)(s) for all z �9 A. In particular, since for z = x(s) the functions x(1 - h~) + x(s)h, converge in F to x as e --+ 0, it follows that r = lim~o0 r - he) + x(s)h~). Therefore r = lim~-~0 r - h,) + x(s)h~)(s) = x(s) , as we claimed. �9

C o r o l l a r y 12. Let F be a strictly convex reflexive Banach space, or F = s or F = CD(X). Then for every r �9 A u t ( B f ) the composition operator Cr is an isolated point in the space o f all composition operators C(H~(BF) , H~~

Proof. First, the identity mapping Id is an isolated point, since it fulfills the assumptions in Propositions 10 and 11, in the three respective cases. The result follows since Cr is a homeomorphism on C(H~~ H~176 BF) ). �9

In closing this section we would like stating the following

C o n j e c t u r e . The identity is an isolated point in C(H~176176 for any Banach

space E.

Support for this conjecture comes from Corollary 8 of [15] and the preceding results.

w P a r t i c u l a r cases In this section we describe the connected components in the space of composition operators for some particular Banach spaces for which we know a formula for the pseudohyperbolic distance. Recall that ([10], Prop. 9) such components always lic inside open balls of radius 2.

Let F be a JB*-triple. For every z , w E F, the Bergman operator B ( z , w ) E E(F) is defined by

B(z , w) = id - 2z <> w + QzQw,

where Qz E s is defined by Q~(w) = { z , w , z } and z o w �9 s is defined by z ~-~ {z, w, x}. For fixed z E Be, we have gz E A ut (B v ) defined by

gz(w) = z + B(z , z)I/2(id + w <> z ) - lw ,

which satisfies gz(O) = z, g'[' = g_= and g-z(z) = 0 (see [13] or [16]). Then

p ( 5 = , a , , , ) = z , , , ,


Indeed, clearly Ib-~(w)ll _< p(*~,*~). For the converse inequality let f E H~176 Ilfll -< 1 and f (w) = O. For any 0 < r < 1, consider r f : BE -r A. Fix z �9 BF and put h := 9-~f(z) o r fo9~ : BF -+ A. Then h(0) = 0, so the Schwarz lemma gives that [h(g_,(w))[ < l[g_~(w)[[. As r --> 1- we conclude that if(z)[ < [Ig-z(W)[[.

L e m m a 13. Let F be a JB*-triple. I f z, w �9 F and ][z][,[[w[[ < r < 1, then

p(6z, 6w) _< ~/1 - (1 - r ) 2 < 1.

Proof. By Proposition 3.1 in [16] with Bz = B(z , z) ~/2,

1 1 -[[g_~(w)[[ 2 - [IB~'l B ( w , z ) B2'[[

< IIBT,~II IIB(w,z)ll l iB;il l .

Since IIB:,~II = ~ 1 and one sees tha t IIB(z,w)fl _< (1 + Ilzll Ilwll) ~ _< (1 + r) ~, we get

1 1

1 - I b _ ~ ( w ) l l = < (1 - r ) ~ "

H e n c e 1 - I t g _ ~ ( ~ ) l l = _> (1 - r ) 2 or I I g - ~ ( ~ ) l l < ~/1 - (1 - r ) 2.

The preceding lemma will be of fundamental use in the following theorem. Note that all that will be really required about the pseudohyperbolic distance is that

sup p(6z,~w) < 1. Ilzll,llwll<<_r

We do not know if this result is true for general Banach spaces.

T h e o r e m 14. I f r r : BE -+ BF, F a JB*- triple, are such that Cr - Cr is a completely continuous operator, then IIG -- C~II < 2.

Proof. We will show that supxeB ~ p(~r fie(z)) < 1. So suppose otherwise and let (xn) �9 BE be a sequence with limn p(6r 6r = 1. Then at least one of the sequences (r or (r does not lie strictly inside By. Indeed, if for some 0 < r < 1, both sequences

were contained in rBF, then by Lemma 13, p(6r 8r _< ~/1 - (1 - r) 2 < 1. Thus we may assume that lim~ [[r = 1. Therefore by the proof of Theorem 10.5 in [2] there is a g �9 H~176 *) which satisfies [g] < 1 and such that (g(r is interpolating for H ~176 where (xk) is a subsequence of (xn). To simplify notation we assume that (xk) is the original sequence (xn). Hence, by Theorem 2.1 in [8] p. 294, there is a sequence (f~) C H ~176 and a constant R > 0 such that

and

f,~(g(r = O, if n ~ k, f,~(g(r = 1,

If,(z)l _< R for all z �9 A. n = l


Thus ( f , ) is equivalent to the sequence of unit vectors (e,) in goo. Since (e,,) is weakly null, f , --> 0 weakly in H~ Let l,~(z) = fn o g(z), z E BF. Then ~, -+ 0 weakly in H~176

Now for each g_r162 E F we choose a functional u , E F* with Ilu,~ll = 1 and un(g-r162 = Iig-r162 and we then set h,~(z) = u , o g_r g,~(z). Thus, h , 6 H~176 with ]]h,t] _< R. Moreover (h,) is a weakly null sequence since given m E M ( H ~ ( B F ) ) , we have

m(h.) = m(~. o g_~(~~ m(~.) -~ O.

Since C~ - C~ is a comple te ly cont inuous operator, the sequence { (C~ - Cr = { ( h . o r - (h~ o r converges to the null function. But

I I (h . o r - (h , o r _> I (h . o r - ( h . o r - -

1~.(o-r162 e , , ( r = p(~r176 6, /~.)) ,

which is a contradiction. �9

I t might be worthwhile to remark at this point that assumption b) in Proposit ion 5 does not imply assumption a): Example 2 in [15] shows a pair of noncompact difference composition operators in H ~' lying at distance less than 2; actually tha t difference cannot be completely continuous (nor weakly compact since H ~176 has Dunford-Pett is property). This follows from the calculations in Example 1.i) of [15] and the fact seen in the above proof tha t lim,~ p(6~(x,), 6r = 0 if l im, llr = 1 and Cr - C r is completely continuous.

Now we present a complete description of the connected components in the space of composition operators, for Hilbert spaces or commutat ive C* algebras. Our pat tern of reasoning follows tha t of [15], and in fact our results exter/d those in [15].

Z--71 In case F is a Hilbert space, for fixed u E BF and z E BF, let m_,~(z) - 1-(z,u}" Then

g_,(z) = (~/1 - Iluil2Q_. + P_, ) (m_ , ( z ) ) ,

where P-u and Q_u are the orthogonal projections along the subspaces generated by - u and its orthogonal, respectively. See also [11].

An explicit expression for the pseudohyperbolic distance can also be given for F : Co(X). Indeed for u,v 6 BE one has

p(~,, s = sup I : _ =~ ,= -~ , I. sEX J. V [ 8 ) U [ 8 )

Let us prove this by using the JB*- triple structure of C0(X), for the product {u, v, f } = uOf. Therefore it is easy to see tha t B(v, v) l /2f = f - v V f . Further we have tha t ( ig+uo(=v) ) - lu = E,,~=o(Ug)"u (uniform limit), so

oo

B(v,v)V=(id + u o ( - , ) ) - X u = E(u9) '~u - ,~ ~-~.(ug)"u. 7 1 = 0 n = O


For each s E X,

co co

9_o(,,)s = - v ( s ) + E ( , , ( s ) , ( ~ ) ) " ~ , ( ~ ) - v(s) ;G-~ ~ ( , , (s> (~ ) )" , , (s ) . n = 0 n = 0

Hence ~,(~)

Ig-, ,(~)sl = I - v(s) + z - ~ ( s > ( s )

and we conclude that

, ( s ) , ( , ) ~ ( ~ ) ~ ( ~ ) - ~(~) 1 - ~(s)u(s) = l i - ~ # - g ) I ,

p(6,,, a,,) = I Ig-, ,(u)l l = s u p l u(s) - v(s)

L e m m a 15. L e t F be a Hi lber t space or F = C o ( X ) . I fO < d < 1, we have

lim p(5., 6,,+,~0,_~,)) = 0 k,i~o

uni formly [or u, z in B F sat i s fy ing p(6~, 5z) <_ d.

Proof. We begin with the Hilbert space case.

m_,,(z)(1 - (z, u>). Therefore,

Since m_~(z ) = z-~ we have z - u = 1-(z,u),

m _ ~ ( u + a(u -- z)) = 1 - ((u + a(u - z ) ) , u}

- a m _ ~ ( z ) ( 1 - {z, u))

1 - (z ,u ) - ( a + 1 ) ( u - - z , u )

- a . L d z ) O - (z, ~) ) -a .L~(z ) ( 1 - ( z , u } ) ( l + ( a + l ) ( r n _ u ( z ) , u ) l + ( a + l ) ( m _ u ( z ) , u ) "

Now bearing in mind that r - 11~II2Q_~ and P-u are linear mappings, we get

9-u(z) g_u(u + a(u - z)) -- - a I + (a + 1)(m_u(z), u)"

Recalling that Q_~ is the orthogonal projection onto the subspace { - u } • it follows that (m_dz), ~) = (P_~(m_.(z)), u) = (o-.(z), ~). Thus

g_.(z) g_~(u + a(u - z)) = - a 1 + (a + 1)(g_u(z) ,u)"

Assuming in addi t ion that (lal + 1)d < 1 so that 1 > (lal + Z) lb-~(~) l l , we have

I Ig - . (u + a(,, - z)) l l =

Ib-~(~/li < i~l ib-~Iz/ii l a [ l l + ( a + l ) I g _ u ( z ) , u ) I - I i _ ( l a l + l ) l ( g _ ~ I z ) , u l l l

Ib-~(z/ll { lg- , , (z>{I < l a l l - 1) < lalz - (lal + 1 ) l b - d = ) l l . I M I - (lal + I b - d z ) l l "

12 A.ron, Galindo, Lindstr6m

The above inequality may be written in terms of the pseudohyperbolic distance as

[alp(6, , 6,) p(~, ~+o(.-=)) <

1 - ([al + 1 ) p ( ~ , ~)'

I-It Now, since 1-(l~l+a)t is increasing as a function of t, it follows that

p(~, ~.+o(~-=)) < 1 - (1~1 + 1)d

from which the result follows. In the case F = Co(X), the fact that p(6~, 6~) < d implies that p(5~(,), 6~(s)) _< d for

all s a X, hence

Md p(6.,,~.+o(,._~)) = sup p(~,,(~), ~.(~)+o(,,_=)(~)) _<

=~x 1 - (1~1 + 1 ) d

Thus the conclusion also holds in this case. �9

T h e o r e m 16. Let F be a Hilbert space or Co(X) and let r r : B E '-+ B y be analytic mappings. The associated composition operators Cr and Cr lie in the same path connected component of C(H~176176176 if and only i f[ICe- Cr < 2.

Proof. We already know that the condition is necessary. So we only prove sufficiency. According to Remark 2, there is d < 1 such that for the pseudohyperbolic distance

p on M(H~176 p(6r 6r < d. Put Ct = tr + (1 - t ) r for each t E [0, 1]. We show that the mapping t E [0, 1] ~-+ Cr G C(H~176 is continuous. By Remark 2 again, it suffices to show that supzEB E P(~r ~r ~ 0 whenever s -+ t. To do this, we first observe that according to [16], Prop. 2.3, the balls'in F of radius less than 1 for the pseudohyperbolic distance arc convex sets. Then since for all x E BE, p(~r162 < d, the convex combination et(x) = tr + (1 - t ) r also satisfies p(6r162 < d. Furthermore, note that whenever t > 0, es(x) = et(x) + ~-~(r - r Hence by Lemma 15 we obtain that lims_.tp(6r162 = 0 uniformly for x E BE. Note that as es(x) = r s ( r r the same argument yields lims-+0 p(6r 6r = 0 uniformly for x E BE. |

By Theorem 14 we get the following result.

C o r o l l a r y 17. Let F be a Hilbert space or Co(X). Ire, %b : BE -+ By are analytic mappings such that Cr - Cr is a completely continuous operator, then Cr and C~ lie in the same path connected component of C(H~176 H~176

Since we do not know whether Theorem 16 and/or Corollary 17 hold for general JB*-triple spaces, we raise the question for the interested reader.


R e f e r e n c e s

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[3] Aron, R. M. , Galindo, P. , and Lindstrbm, M. , Compact homomorphisms between algebras of analytic functions, Studia Math. 123 (3) (1997), 235-247.

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[9] Galindo, P. , Lindstrbm, M. and Ryan, R. , Weakly compact composition operators between algebras of bounded analytic functions, Proc. Amer. Math. Soc. 128 (1) (2000), 149-155.

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[11] Goebel, K. and Reich, S. , Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker Inc. (1984).

[12] Isidro, J. M. and Kaup, W. , Weak continuity of holomorphic automorphisms in JB* triples, Math. Z. 210 (2) (1992), 277-288.

[13] Kaup, W. , A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), 503-529.

[14] Kbnig, H. , Zur abstrakten Theorie der analytischen Funktionen II, Math. Ann. 163 (1966), 9-17.

[15] MacClucr, B . , Ohno, S. and Zhao, 1~., Topological structure of the space of composition operators on H ~176 Int. Eq. Op. Theory 40 (2001),481-494.

[16] Mellon, P. , Holomorphie invariance on bounded symmetric domains, J. reine angew. Math. 523 (2000), 199-223.

[17] Mujica, J. , Complex analysis in Banach spaces, North Holland (1986).

[18] Rudin, W. , Function Theory on Polydiscs, Benjamin (1969).

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Department of Mathematics Kent State University Kent, 0hio,44242 USA e-mail: [email protected]

Departamento de An~lisis Matem~tico Universidad de Valencia 46100 Burjasot,Valencia, Spain e-mail: [email protected]

Department of Mathematics /~bo Akademi University FIN-20500/~bo, Finland c-marl: [email protected]

2000 AMS Classification Numbers: Primary 46J15. Secondary 46E15, 46G20

Submitted: November 4, 2001 Revised: January 9, 2002

Connected components in the space of composition operators onH∞ functions of many variables

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