Bilinear Regular Operators on Quasi-Banach Lattices and ...websites.math.leidenuniv.nl/.../brandanidasilva.pdf · Multilinear operators arise naturally in many areas of classical

Bilinear Regular Operators onQuasi-Banach Lattices and

CompactnessEduardo Brandani da SilvaMaringa State University - Brazil

joint work with Dicesar L. Fernandez

Bilinear Regular Operators on Quasi-Banach Lattices and Compactness – p. 1/46

- Introduction - Overview

Multilinear operators arise naturally in many areasof classical and harmonic analysis, as well asfunctional analysis, including the theory of Banachoperator ideals.



Multilinear operators arise naturally in many areasof classical and harmonic analysis, as well asfunctional analysis, including the theory of Banachoperator ideals.

Fundamental multilinear operators arising inharmonic analysis include convolutions,paraproducts and multilinear Fourier multiplieroperators.



In the last years, several singular multilinearoperators have been intensively studied and theresearch on bilinear Hilbert transform, originated bythe work of M. Lacey and C. Thiele




"On Calderón’s conjecture", Ann. Math. 149 (1999)475–496




"On Calderón’s conjecture", Ann. Math. 149 (1999)475–496

has shown the need for the development of asystematic analysis of bilinear operators.



In the following paper we may find more detailsabout this subject.



In the following paper we may find more detailsabout this subject.

L. Grafakos and N.J. Kalton,The Marcinkiewiczmultiplier condition for Bilinear operators, StudiaMath. 146 (2001), pp. 115–156.



Another topic is the theory of operator ideals ands-numbers in Banach spaces.




For the multilinear case, a systematic theory appearsin




For the multilinear case, a systematic theory appearsin

D. L. Fernandez, M. Mastylo and E. B. da Silva.Quasi s-numbers and measures of non-compactnessof multilinear operators. Ann. Acad. Scient.Fennicae Math. 38, (2013), 805-823.



The study of multilinear operators in interpolation ofBanach spaces is an old topic and severalresearchers worked about. See, for instance:




M. Mastylo. On interpolation of bilinear operators.J. Func. Anal. 214 (2004) 260-283.




M. Mastylo. On interpolation of bilinear operators.J. Func. Anal. 214 (2004) 260-283.

D. L. Fernandez and E. B. Silva.Interpolation ofbilinear operators and compactness. NonlinearAnal. 73 (2010) 526-537



Also, connections between positive bilinearoperators, functionals spaces and interpolationtheory were explored in




A.G. Kusraev,Jensen type inequalities for positivebilinear operators, Positivity 16 (2012),pp.131–142.




A.G. Kusraev,Jensen type inequalities for positivebilinear operators, Positivity 16 (2012),pp.131–142.

L. Maligranda,Positive bilinear operators inCalderon-Lozanovski spaces, Arch. Math. 81(1)(2003), pp. 26–37.



Some results for interpolation of bilinear operatorshave been applied in general theory of Banachspaces and in the theory of multilinearp-summingoperators. See, respectively




G. Pisier, "A remark onπ2(ℓ2, ℓ2)", Math.Nachrichten 148 (1990) 243–245,




G. Pisier, "A remark onπ2(ℓ2, ℓ2)", Math.Nachrichten 148 (1990) 243–245,

G. Botelho, C. Michels and D. Pellegrino, "Complexinterpolation and summability properties ofmultilinear operators", Rev. Mat. Complut. 23(2010) 139–161.



But, not all results for the linear case are generalizedto the bilinear case.




For instance, the linear Marcinkiewicz multipliertheorem, whose natural bilinear version fails, asshown by L. Grafakos and N. Kalton in




For instance, the linear Marcinkiewicz multipliertheorem, whose natural bilinear version fails, asshown by L. Grafakos and N. Kalton in

"The Marcinkiewicz multiplier condition for bilinearoperators", Studia Math. 146 (2001) 115–156.



In the paper



In the paper

Q. Bu, G. Buskes and A. G. Kusraev,Bilinear Mapson Product of Vector lattices: A Survey, Positivity-Trends in Mathematics, 97-128, 2007.



In the paper

Q. Bu, G. Buskes and A. G. Kusraev,Bilinear Mapson Product of Vector lattices: A Survey, Positivity-Trends in Mathematics, 97-128, 2007.

it is presented several important results aboutbilinear maps on products of normed vector lattices.



Quasi-Banach spaces and their relations with severalbranches of mathematics is a subject which has beenlately interested to many researchers.



Quasi-Banach spaces and their relations with severalbranches of mathematics is a subject which has beenlately interested to many researchers.

Besides the classical works by Aoki, Rolewicz andKalton, the studies of analytic and geometric aspectsare one of the main issues for these spaces, withmany results obtained recently.



For instance, several results for interpolation ofbilinear operators in quasi-Banach spaces wereobtained in



For instance, several results for interpolation ofbilinear operators in quasi-Banach spaces wereobtained in

L. Grafakos and M. Mastylo.Interpolation ofbilinear operators between quasi-Banach spaces.Positivity 10 (2006), 409–429.


- Introduction

In the current work, positive and regular bilinearoperators on quasi-normed functional spaces aredefined and some properties and characterizationson lattices and quasi-normed lattices are obtained.


- Introduction

In the current work, positive and regular bilinearoperators on quasi-normed functional spaces aredefined and some properties and characterizationson lattices and quasi-normed lattices are obtained.

We introduce a variant definition of functionalquasi-norm and prove some theorems characterizingcompactness of bilinear operators.


- Introduction

Using a very interesting and powerful definition ofadjoint of a bilinear mapping, introduced in


- Introduction


M.S Ramanujan and E. Schock,Operator ideals andspaces of bilinear operators, Linear and Mult. Alg.18(4) (1985), pp. 307–318 ,


- Introduction


M.S Ramanujan and E. Schock,Operator ideals andspaces of bilinear operators, Linear and Mult. Alg.18(4) (1985), pp. 307–318 ,

relations between positive and regular bilinearoperators and their adjoint on normed functionalspaces are proved.


- Introduction

We call the attention that Ramanujan-Schock’sdefinition differs from the more known definition ofadjoint of a bilinear map, introduced by Arens in


- Introduction

We call the attention that Ramanujan-Schock’sdefinition differs from the more known definition ofadjoint of a bilinear map, introduced by Arens in

R. Arens,The adjoint of a bilinear operation. Proc.Amer. Math. Soc. 2, pp. 839-848, (1951).


- Positive Operators

In what follows, the first definitions and results arepresented. For a vector latticeX, the positive cone isdenoted byX+.



In what follows, the first definitions and results arepresented. For a vector latticeX, the positive cone isdenoted byX+.

Definition 1. LetX, Y andZ be vector lattices. Abilinear operatorT : X × Y → Z is positive if givenx ∈ X+ andy ∈ Y+, one hasT (x, y) ∈ Z+.



Proposition 1. LetX, Y andZ be vector latticesandT : X × Y → Z a bilinear positive operator.Then,

|T (x, y)| ≤ T (|x|, |y|),

for all (x, y) ∈ X × Y .



Proposition 1. LetX, Y andZ be vector latticesandT : X × Y → Z a bilinear positive operator.Then,

|T (x, y)| ≤ T (|x|, |y|),


Definition 2. LetX, Y andZ be ordered vectorspaces. A bilinear operatorT : X × Y → Z isregular if it may be written as

T = T1 − T2,

whereT1 andT2 are positive bilinear operators.Bilinear Regular Operators on Quasi-Banach Lattices and Compactness – p. 17/46


Theorem 1. LetX, Y andZ be vector lattices. Abilinear operatorT : X × Y → Z is regular if, andonly if, there exists a positive bilinear operatorS : X × Y → Z such that

|T (x, y)| ≤ S(|x|, |y|),

for all (x, y) ∈ X × Y . The operatorS is called apositive upper bound of the operatorT .



Theorem 1. LetX, Y andZ be vector lattices. Abilinear operatorT : X × Y → Z is regular if, andonly if, there exists a positive bilinear operatorS : X × Y → Z such that

|T (x, y)| ≤ S(|x|, |y|),

for all (x, y) ∈ X × Y . The operatorS is called apositive upper bound of the operatorT .

Definition 3. An ordered setX is Dedekindcomplete if every non-empty subset ofX that isbounded above admits a supremum (inX).



Theorem 2. LetX andY be vector lattices andZ aDedekind complete vector lattice. A bilinearoperatorT : X × Y → Z is regular if, and only if,for each(u, v) ∈ X+ × Y+, there existsω ∈ Z+ suchthat

T (x, y) ≤ ω,

for all (x, y) ∈ X+ × Y+ with 0 ≤ x ≤ u and0 ≤ y ≤ v.


- Quasi-Normed Lattices

Definition 4. A quasi-norm in a vector spaceX isan application||.|| of X in [0,∞[ such that, forx, y ∈ X andλ ∈ R, verifies the conditions



Definition 4. A quasi-norm in a vector spaceX isan application||.|| of X in [0,∞[ such that, forx, y ∈ X andλ ∈ R, verifies the conditions

QN1) ||x|| = 0 ⇐⇒ x = 0;

QN2) ||λx|| = |λ| ||x||;

QN3) ||x+ y|| ≤ C (||x||+ ||y||),

for someC ≥ 1.



A vector spaceX endowed with a quasi-norm iscalled aquasi-normed space.



A vector spaceX endowed with a quasi-norm iscalled aquasi-normed space.

A quasi-Banach spaceis a quasi-normed spacewhich is complete in the topology generated by

d(x, y) = ||x− y||.



A classic result is the Aoki-Rolewicz’s Theorem:



A classic result is the Aoki-Rolewicz’s Theorem:

Theorem 3. If X is a quasi-normed space endowedwith a quasi-norm||.||, there exists a constantα,0 < α ≤ 1, and an equivalent quasi-norm|||.||| suchthat

|||x+ y|||α ≤ |||x|||α + |||y|||α

for all x, y in X.



Definition 5. If a quasi-normed space is also avector lattice(X,≤), we sayX is aquasi-normedlattice if

|x| ≤ |y| =⇒ ||x|| ≤ ||y||.



Definition 5. If a quasi-normed space is also avector lattice(X,≤), we sayX is aquasi-normedlattice if

|x| ≤ |y| =⇒ ||x|| ≤ ||y||.

Besides, if a quasi-normed lattice is complete, wesay it is aquasi-Banach lattice.



Theorem 4. LetX andY be quasi-Banach latticesandZ a quasi-normed lattice. If a bilinear operatorT : X × Y → Z is positive, then it is bounded.



Theorem 4. LetX andY be quasi-Banach latticesandZ a quasi-normed lattice. If a bilinear operatorT : X × Y → Z is positive, then it is bounded.

Corollary 1 . In the conditions of the Theorem 4, ifthe operatorT : X × Y → Z is regular, it is alsobounded.


- Operators and Functionals Spaces

We introduce now the function spaces which we willdeal with. We define a variant general concept offunctional quasi-norm which allow us to generalizeseveral functional spaces.



We introduce now the function spaces which we willdeal with. We define a variant general concept offunctional quasi-norm which allow us to generalizeseveral functional spaces.

Let (Ω, µ) a measure space. We denote byL0+ = L0

+(Ω, µ) the cone of realµ-measurable, nonnegative andµ-a.e. finite functions onΩ.



Definition 6. An applicationρ : L0+ → [0,∞] is a

functional quasi-norm if, for all f, g ∈ L0+, for all

λ > 0 and for all subsetD ⊂ Ω, with µ(D) < ∞,the following conditions are verified:






C1) ρ(f) = 0 ⇐⇒ f = 0, µ− a.e.;






C1) ρ(f) = 0 ⇐⇒ f = 0, µ− a.e.;

C2) ρ(λf) = λρ(f) for all λ > 0;






C1) ρ(f) = 0 ⇐⇒ f = 0, µ− a.e.;

C2) ρ(λf) = λρ(f) for all λ > 0;

C3) ρ(f + g) ≤ C (ρ(f) + ρ(g)),for some C ≥ 1.



C4) 0 ≤ g ≤ f µ− a.e. =⇒ ρ(g) ≤ ρ(f);



C4) 0 ≤ g ≤ f µ− a.e. =⇒ ρ(g) ≤ ρ(f);

C5) ρ(χD) < ∞;



C4) 0 ≤ g ≤ f µ− a.e. =⇒ ρ(g) ≤ ρ(f);

C5) ρ(χD) < ∞;

C6) λµ(x ∈ D ; |f(x)| ≥ λ)1/p ≤ C ′ ρ(f),

for somep > 0 and constantC ′ > 0, dependent ofDandρ, and independent off .



The spaceL∞ = L∞(Ω, µ) is defined as the set ofall measurable real functions onΩ, which areessentially bounded, i.e. bounded up to a set ofmeasure zero. Forf ∈ L∞, its norm is given by:

||f || = infa ∈ R : µ(t : f(t) > a) = 0 .



We denote byS = S(Ω, µ) the subclass of simplefunctions.



We denote byS = S(Ω, µ) the subclass of simplefunctions.

Definition 7. Let ρ be a functional quasi-norm inL0+(Ω, µ). The class of the functionsf ∈ L0 such

thatρ(|f |) < ∞ is denoted byX = X(Ω, µ, ρ).



Theorem 5. Let ρ be a functional norm andX = X(Ω, µ, ρ). Forf ∈ X let

||f ||X = ρ(|f |).

Then,X is a quasi-normed vector subspaceverifying the inclusions

S ⊂ X → L0 .



Definition 8. LetX = X(Ω, µ, ρ) a quasi-normedfunctional space. A functionf ∈ X hasabsolutelycontinuous quasi-normif, given ε > 0 there existsδ > 0 such that,µ(D) < δ implies

||fχD|| < ε.




||fχD|| < ε.

We denote byXa the subspace ofX of all absolutelycontinuous quasi-normed functions .




||fχD|| < ε.

We denote byXa the subspace ofX of all absolutelycontinuous quasi-normed functions .

X has absolutely continuous quasi-norms ifX = Xa.



Definition 9. A family M ⊂ X hasequi-absolutelycontinuous quasi-normif, for all ε > 0 there existsδ > 0 such thatµ(D) < δ implies

||PDf || < ε,

for all f ∈ M, wherePDf(s) = f(s) if s ∈ D andPDf(s) = 0 if s /∈ D.



Definition 9. A family M ⊂ X hasequi-absolutelycontinuous quasi-normif, for all ε > 0 there existsδ > 0 such thatµ(D) < δ implies

||PDf || < ε,

for all f ∈ M, wherePDf(s) = f(s) if s ∈ D andPDf(s) = 0 if s /∈ D.

Theorem 6. LetX = X(Ω, µ, ρ1) andY = Y (Ω, ν, ρ2) be functional quasi-normed spaces.Each bounded bilinear operatorT acting fromX × Y toL∞ is regular.


- Compactness Theorems

In what follows we give some characterizations ofcompact bilinear operators on the quasi-normedfunctional spaces.




Let X = X(Ω1, µ, ρ1), Y = Y (Ω2, ν, ρ2) andZ = Z(Ω3, υ, ρ3) be quasi-normed functionalspaces.




Let X = X(Ω1, µ, ρ1), Y = Y (Ω2, ν, ρ2) andZ = Z(Ω3, υ, ρ3) be quasi-normed functionalspaces.

We denote byBil(X × Y, Z) the family of allbounded bilinear operators fromX × Y toZ.



Definition 10. A bounded bilinear operatorT : X × Y → Z is compact in measure if the imageT (un, vn), of any bounded sequence(un, vn) ofX × Y , contains a Cauchy subsequence in respect tothe measureυ, that is,




if max‖un‖X , ‖vn‖Y ≤ C, then there exists asubsequence(unk

, vnk) such that, givenε > 0 and

δ > 0, there existsN = N(ε, δ) with




if max‖un‖X , ‖vn‖Y ≤ C, then there exists asubsequence(unk

, vnk) such that, givenε > 0 and

δ > 0, there existsN = N(ε, δ) with

υ(s ∈ Ω3 : |T (unk, vnk

)(s)− T (umk, vmk

)(s)| >ε) < δfor all nk,mk > N .



Theorem 7. LetX andY quasi-normed functionalspaces and suppose thatZ has absolutely continuousquasi-norms, i.eZ = Za. LetT : X × Y → Z be abounded bilinear operator.




Then,T is compact if, and only if,T is compact inmeasure and the functions in the set




Then,T is compact if, and only if,T is compact inmeasure and the functions in the set

T (f, g) : ||f ||X ≤ 1 , ||g||Y ≤ 1

have equi-absolutely continuous quasi-norms.



Theorem 8. LetX, Y andZ be quasi-normedfunctional spaces. Moreover, suppose thatZ hasabsolutely continuous quasi-norms, i.eZ = Za, andυ(Ω3) < ∞.




A bilinear bounded operatorT : X × Y → Z iscompact if, and only if,T is compact in measure andsatisfies




A bilinear bounded operatorT : X × Y → Z iscompact if, and only if,T is compact in measure andsatisfies

limυ(E)→0

‖PET‖Bil(X×Y,Z) = 0 ,

whereE ⊂ Z.



Theorem 9. LetX, Y andZ be quasi-normedfunctional spaces, whereµ(Ω1) < ∞, ν(Ω2) < ∞,υ(Ω3) < ∞ andZ has absolutely continuousquasi-norms, i.eZ = Za.




A bilinear regular operatorT : X × Y → Z iscompact if, and only if,T is compact in measure andsatisfies




A bilinear regular operatorT : X × Y → Z iscompact if, and only if,T is compact in measure andsatisfies

limυ(E)+µ(D1)+ν(D2)→0

‖PET (PD1, PD2

)‖Bil(X×Y,Z) = 0 ,

whereD1 ⊂ Ω1, D2 ⊂ Ω2 andE ⊂ Ω3.


- Compactness and adjoint operators

The present results are devoted to the relationshipsamong the corresponding regular bilinear operatorsand their adjoints.



The present results are devoted to the relationshipsamong the corresponding regular bilinear operatorsand their adjoints.

Let us recall that Schauder’s well-known resultstates that an operatorT between Banach spaces iscompact if, and only if, its adjoint,T ∗, is compact.



Ramanujan and Schock studied in

M.S Ramanujan and E. Schock.Operator ideals andspaces of bilinear operators. Linear and Mult. Alg.18(4) (1985), pp. 307–318.



Ramanujan and Schock studied in

M.S Ramanujan and E. Schock.Operator ideals andspaces of bilinear operators. Linear and Mult. Alg.18(4) (1985), pp. 307–318.

ideals of bilinear operators between Banach spaces,including the ideal of bilinear compact operators,i.e.,T ∈ Bill(X × Y, Z) such thatT (UX × UY ) isrelatively compact inZ.



Definition 11. GivenT ∈ Bil(X × Y, Z), theadjoint ofT is the linear mapT× : Z∗ → Bil(X × Y ) is given by

T×z∗(x, y) = z∗(T (x, y)), (x, y) ∈ X × Y.




T×z∗(x, y) = z∗(T (x, y)), (x, y) ∈ X × Y.

T× is a bounded operator.




T×z∗(x, y) = z∗(T (x, y)), (x, y) ∈ X × Y.

T× is a bounded operator.

‖T‖ = ‖T×‖.



ForT× may be proved the analogue of Schauder’stheorem which states that ifT ∈ Bil(X × Y, Z), thenT is compact if, and onlyif T× is compact.



ForT× may be proved the analogue of Schauder’stheorem which states that ifT ∈ Bil(X × Y, Z), thenT is compact if, and onlyif T× is compact.

And more, ifT ∈ Bil(X × Y, Z) andS ∈ L(Z,W ),then

(ST )× = T×S∗

whereS∗ is the classical linear adjoint.



From now, we are assuming thatZ = Lp(Ω3, υ)with 1 < p < ∞.




Theorem 10. LetX andY be normed functionalspaces, whereµ(Ω1) < ∞ andν(Ω2) < ∞,υ(Ω3) < ∞.





Then, a bilinear bounded operatorT : X × Y → Zis compact if, and only if,(T×)∗ is compact inmeasure and satisfies





Then, a bilinear bounded operatorT : X × Y → Zis compact if, and only if,(T×)∗ is compact inmeasure and satisfies

limµ(E)→0

‖T×P ∗E‖ = 0 ,

whereE ⊂ Ω3.Bilinear Regular Operators on Quasi-Banach Lattices and Compactness – p. 42/46


In what follows, letΩ1 = Ω2 = Ω andµ = ν.




Definition 12. GivenD ⊂ Ω, we define

PD : Bil(X × Y ) → Bil(X × Y )




Definition 12. GivenD ⊂ Ω, we define

PD : Bil(X × Y ) → Bil(X × Y )

such that, forb ∈ Bil(X × Y ) thenPD(b) ∈ Bil(X × Y ) and

PD(b)(x, y) = b(PDx, PDy)




Proposition 2. PD : Bil(X × Y ) → Bil(X × Y ) isa bounded linear operator.




Proposition 3. Considering the sequence ofoperators

Z ′ P ∗

E−→ Z ′ T×

−→ Bil(X × Y )PD−→ Bil(X × Y ) ,




Proposition 3. Considering the sequence ofoperators

Z ′ P ∗

E−→ Z ′ T×

−→ Bil(X × Y )PD−→ Bil(X × Y ) ,

one has

‖PDT×P ∗

E‖L(Z ′,Bil(X×Y )) = ‖PET (PD, PD)‖Bil(X×Y,Z) .



Theorem 11. LetX andY be normed functionalspaces, whereµ(Ω) < ∞ andυ(Ω3) < ∞.




A bounded bilinear regular operatorT : X × Y → Z is compact if, and only if,(T×)∗ is compact in measure and satisfies




A bounded bilinear regular operatorT : X × Y → Z is compact if, and only if,(T×)∗ is compact in measure and satisfies

limυ(E)+µ(D)→0

‖PDT×P ∗

E‖L(Z ′,Bil(X×Y )) =

limυ(E)+µ(D)→0

‖PET (PD, PD)‖Bil(X×Y,Z) = 0 .



Thank you for your attention!


Bilinear Regular Operators on Quasi-Banach Lattices and ...websites.math.leidenuniv.nl/.../brandanidasilva.pdf · Multilinear operators arise naturally in many areas of classical

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