p-potencias y compacidad para operadores entre espacios de banach de funciones

p-potencias y compacidad para operadores entre espacios deBanach de funciones

Enrique A. Sanchez Perez

I.U.M.P.A.-U. Politecnica de Valencia,Joint work with P. Rueda (Universidad de Valencia)

Murcia 2012

E. Sanchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 1/14

Resumen: En general, la compacidad de un operador entre espacios de Banach defunciones y el hecho de que el conjunto de las imagenes de la funciones caracterısticasde conjuntos medibles sea relativamente compacto, son propiedades diferentes. Enesta charla explicaremos como se puede caracterizar la diferencia entre estaspropiedades en terminos de las p-potencias del espacio de Banach de funciones queconstituye el dominio del operador. Como aplicacion, veremos cuando se puedeextender un operador a un espacio L1 preservando la compacidad.

(Ω,Σ, µ) finite measure space.

X (µ) Banach function space over µ (Banach ideal of measurable functions).

T : X (µ)→ E linear and continuous operator, E Banach space.

The essential range of an operator T : X (µ)→ E is the set T (χA) : A ∈ Σ.

A continuous operator T : X (µ)→ E is said to be essentially compact if itsessential range is relatively compact in E .

Let 0 < p <∞. If X (µ) is a Banach function space, the p-th power of X is defined as

X[p] := f ∈ L0(µ) : |f |1/p ∈ X.

It is a quasi-Banach function space over µ when endowed with the seminorm‖f ‖X[p]

:= ‖|f |1/p‖pX , f ∈ X[p].

It is a Banach space and the above expression defines a norm if and only if X isp-convex with p-convexity constant 1. X[1/p] is sometimes called the p-convexificationof X .

For instance, Lp [0, 1][p] = L1[0, 1] and L1[0, 1][1/p] = Lp [0, 1] isometrically.

X[p] := f ∈ L0(µ) : |f |1/p ∈ X.

:= ‖|f |1/p‖pX , f ∈ X[p].

X[p] := f ∈ L0(µ) : |f |1/p ∈ X.

:= ‖|f |1/p‖pX , f ∈ X[p].

X[p] := f ∈ L0(µ) : |f |1/p ∈ X.

:= ‖|f |1/p‖pX , f ∈ X[p].

Compactness and essential compactness are not equivalent properties for operators inBanach function spaces.

Example: Volterra operator. Let 1 < r <∞. Consider the Volterra operatorVr : Lr [0, 1]→ Lr [0, 1] given by Vr (f )(x) :=

∫ x0 f (t)dt, x ∈ [0, 1], f ∈ Lr [0, 1]. It is

compact.

The integration map Iνr : L1(νr )→ Lr [0, 1] associated to the Volterra measureνr : B([0, 1])→ Lr [0, 1], νr (A) := Vr (χA), A ∈ B([0, 1]), is not compact.

BUT: it is essentially compact as a consequence of Vr being compact.

∫ x0 f (t)dt, x ∈ [0, 1], f ∈ Lr [0, 1]. It is

compact.

∫ x0 f (t)dt, x ∈ [0, 1], f ∈ Lr [0, 1]. It is

compact.

∫ x0 f (t)dt, x ∈ [0, 1], f ∈ Lr [0, 1]. It is

compact.

Theorem: Let 1 ≤ p <∞. Let T : X → (E , τ) be a continuous operator and let U bea basis of absolutely convex open neighborhoods of 0 in (E , τ). Consider the followingassertions:

(a) The operator T : X → (E , τ) is compact.

(b) For each g ∈ BX[1/p′ ]the image T (gBX[1/p]

) is relatively τ -compact in E and for

every U ∈ U there exists gU ∈ X[1/p′] such that T (BX ) ⊂ T (gUBX[1/p]) + U.

(c) For each g ∈ BX[1/p′ ]the image T (gBX[1/p]

) is relatively τ -compact in E and for

every U ∈ U there exists KU > 0 such that T (BX ) ⊂ T (KUBX[1/p]) + U.

Then (a) implies (b) and (c). If besides τ is metrizable then (b) or (c) implies (a).

Lemma: Let 1 < p <∞. Let X (µ) be an order continuous Banach function spaceand E be a Banach space. A continuous operator T : X (µ)→ E is essentiallycompact if and only if for every h ∈ X[1/p′] the map Th : X[1/p] → E given byTh(·) := T (h ·), is compact.

Corollary: Let 1 < p <∞. Let X (µ) be an order continuous Banach function spaceand E be a Banach space. If T : X (µ)→ E is a continuous essentially compactoperator then the restriction T |X[1/p]

: X[1/p] → E is compact.

Lemma: Let 1 < p <∞. Let X (µ) be an order continuous Banach function spaceand E be a Banach space. A continuous operator T : X (µ)→ E is essentiallycompact if and only if for every h ∈ X[1/p′] the map Th : X[1/p] → E given byTh(·) := T (h ·), is compact.

Corollary: Let 1 < p <∞. Let X (µ) be an order continuous Banach function spaceand E be a Banach space. If T : X (µ)→ E is a continuous essentially compactoperator then the restriction T |X[1/p]

: X[1/p] → E is compact.

Theorem: Let 1 ≤ p <∞ and let X (µ) be an order continuous Banach functionspace. The following statements for a continuous operator T : X → E are equivalent:

(i) T is compact.

(ii) T is essentially compact and for every ε > 0 there exists hε ∈ X[1/p′] such thatT (BX ) ⊂ T (hεBX[1/p]

) + εBE .

(iii) T is essentially compact and for every ε > 0 there exists Kε > 0 such thatT (BX ) ⊂ T (KεBX[1/p]

) + εBE .

KERNEL OPERATORS

Corollary:Let X (µ), F and T as above. Suppose also that T is essentially compact.Then

limK→∞

supf∈BX

∥∥T (f χ|f |>K)‖ = 0

implies that T is compact.

Let Tk be a positive kernel operator given by

(Tk f )(y) :=

∫[0,1]

f (x)k(y , x) dx .

Corollary: Let T be an essentially compact (positive) kernel operatorT : X (µ)→ Y (ν) such that the kernel k satisfies that

limµ(A)→0

∥∥∥‖χAk(x , y)‖X ′ (y)∥∥∥Y

Then T is compact.

KERNEL OPERATORS

limK→∞

supf∈BX

∥∥T (f χ|f |>K)‖ = 0

(Tk f )(y) :=

∫[0,1]

f (x)k(y , x) dx .

limµ(A)→0

∥∥∥‖χAk(x , y)‖X ′ (y)∥∥∥Y

Then T is compact.

KERNEL OPERATORS

limK→∞

supf∈BX

∥∥T (f χ|f |>K)‖ = 0

(Tk f )(y) :=

∫[0,1]

f (x)k(y , x) dx .

limµ(A)→0

∥∥∥‖χAk(x , y)‖X ′ (y)∥∥∥Y

Then T is compact.

KERNEL OPERATORS

limK→∞

supf∈BX

∥∥T (f χ|f |>K)‖ = 0

(Tk f )(y) :=

∫[0,1]

f (x)k(y , x) dx .

limµ(A)→0

∥∥∥‖χAk(x , y)‖X ′ (y)∥∥∥Y

Then T is compact.

COMPACT OPTIMAL DOMAINS FOR ESSENTIALLY COMPACT OPERATORS

Characterize when the optimal domain of an operator is an L1-space. Consider avector measure m : Σ→ E and let Im : L1(m)→ E be the integration operatorIm(f ) =

∫Ω fd m, f ∈ L1(m).

Proposition:Let 1 ≤ p <∞. The following are equivalent:

(a) The integration operator Im : L1(m)→ (E , ‖ ‖E ) is compact.

(b) R(m) is relatively compact and for every ε > 0 there exists gε ∈ Lp′(m) such

that Im(BL1(m)) ⊂ Im(gεBLp(m)) + εBE .

(c) R(m) is relatively compact and for every ε > 0 there exists Kε > 0 such thatIm(BL1(m)) ⊂ Im(KεBLp(m)) + εBE .

∫Ω fd m, f ∈ L1(m).

Corollary:Let m : Σ→ E be a vector measure with relatively compact range. Supposethat the integration map Im : L1(m)→ E satisfies an 1/p-th power approximation.Then L1(m) = L1(|m|).

When an operator T has optimal extensions that satisfy 1/p-th power approximations?

Corollary:Let m : Σ→ E be a vector measure with relatively compact range. Supposethat the integration map Im : L1(m)→ E satisfies an 1/p-th power approximation.Then L1(m) = L1(|m|).

When an operator T has optimal extensions that satisfy 1/p-th power approximations?

Theorem: Let 1 < p <∞, let X (µ) be an order continuous Banach function space, Ea Banach space and T : X (µ)→ E a continuous operator. The following statementsare equivalent for T .

(i) T is essentially compact, and for each ε > 0 there is hε ∈ X[1/p′] such that

supf∈B

L1(mT )∩X (µ)

g∈BLp (mT )∩X[1/p]

(∥∥T (f − hεg)∥∥E

))< ε.

(ii) T is essentially compact, and for each ε > 0 there is a constant Kε > 0 such that

supf∈B

L1(mT )∩X (µ)

g∈BLp (mT )∩X[1/p]

(∥∥T (f − Kεg)∥∥E

))< ε.

(iii) The optimal domain of T is L1(|mT |) and the extension ImT is compact, i.e. Tfactorizes compactly as

Corollary: Let 1 < p <∞, let X (µ) be an order continuous Banach function space, Ea Banach space and T : X (µ)→ E a µ-determined essentially compact continuousoperator. The following statements are equivalent.

(i) Each extension of T to an order continuous Banach function space satisfies a1/p-th power approximation.

(ii) ImT satisfies a 1/p-th power approximation.

(iii) T admits a maximal extension that satisfies a 1/p-th power approximation.

J. M. Calabuig, O. Delgado and E. A. Sanchez Perez, Factorizing operators onBanach function spaces through spaces of multiplication operators, J. Math.Anal. Appl. 364 (2010), 88-103.

I. Ferrando and J. Rodrıguez, The weak topology on Lp of a vector measure,Topology Appl. 155(13) (2008) , 1439–1444.

S. Okada, Does a compact operator admit a maximal domain for its compactlinear extension?, in: Operator Theory: Advances and Applications, 201.Birkhauser Verlag, Basel, 2009, 313–322.

A. R. Schep, When is the optimal domain of a positive linear operator a weightedL1-space?, in: Operator Theory: Advances and Applications, 201. BirkhauserVerlag, Basel, 2009, 361-369.

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