1. p-potencias y compacidad para operadores entre espacios de
Banach de funciones
Enrique A. S´anchez P´erez
I.U.M.P.A.-U. Polit´ecnica de Valencia,
Joint work with P. Rueda (Universidad de Valencia)
Murcia 2012
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 1/14

2. Resumen: En general, la compacidad de un operador entre espacios de Banach de
funciones y el hecho de que el conjunto de las im´agenes de la funciones caracter´ısticas
de conjuntos medibles sea relativamente compacto, son propiedades diferentes. En
esta charla explicaremos como se puede caracterizar la diferencia entre estas
propiedades en t´erminos de las p-potencias del espacio de Banach de funciones que
constituye el dominio del operador. Como aplicaci´on, veremos cu´ando se puede
extender un operador a un espacio L1 preservando la compacidad.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 2/14

3. (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 its
essential range is relatively compact in E.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 3/14

4. (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 its
essential range is relatively compact in E.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 3/14

5. (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 its
essential range is relatively compact in E.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 3/14

6. (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 its
essential range is relatively compact in E.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 3/14

7. (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 its
essential range is relatively compact in E.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 3/14

8. 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 p
X , f ∈ X[p].
It is a Banach space and the above expression defines a norm if and only if X is
p-convex with p-convexity constant 1. X[1/p] is sometimes called the p-convexification
of X.
For instance, Lp[0, 1][p] = L1[0, 1] and L1[0, 1][1/p] = Lp[0, 1] isometrically.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 4/14

9. 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 p
X , f ∈ X[p].
It is a Banach space and the above expression defines a norm if and only if X is
p-convex with p-convexity constant 1. X[1/p] is sometimes called the p-convexification
of X.
For instance, Lp[0, 1][p] = L1[0, 1] and L1[0, 1][1/p] = Lp[0, 1] isometrically.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 4/14

10. 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 p
X , f ∈ X[p].
It is a Banach space and the above expression defines a norm if and only if X is
p-convex with p-convexity constant 1. X[1/p] is sometimes called the p-convexification
of X.
For instance, Lp[0, 1][p] = L1[0, 1] and L1[0, 1][1/p] = Lp[0, 1] isometrically.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 4/14

11. 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 p
X , f ∈ X[p].
It is a Banach space and the above expression defines a norm if and only if X is
p-convex with p-convexity constant 1. X[1/p] is sometimes called the p-convexification
of X.
For instance, Lp[0, 1][p] = L1[0, 1] and L1[0, 1][1/p] = Lp[0, 1] isometrically.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 4/14

12. Compactness and essential compactness are not equivalent properties for operators in
Banach function spaces.
Example: Volterra operator. Let 1 < r < ∞. Consider the Volterra operator
Vr : Lr [0, 1] → Lr [0, 1] given by Vr (f )(x) := x
0 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 measure
νr : 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.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 5/14

13. Compactness and essential compactness are not equivalent properties for operators in
Banach function spaces.
Example: Volterra operator. Let 1 < r < ∞. Consider the Volterra operator
Vr : Lr [0, 1] → Lr [0, 1] given by Vr (f )(x) := x
0 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 measure
νr : 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.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 5/14

14. Compactness and essential compactness are not equivalent properties for operators in
Banach function spaces.
Example: Volterra operator. Let 1 < r < ∞. Consider the Volterra operator
Vr : Lr [0, 1] → Lr [0, 1] given by Vr (f )(x) := x
0 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 measure
νr : 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.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 5/14

15. Compactness and essential compactness are not equivalent properties for operators in
Banach function spaces.
Example: Volterra operator. Let 1 < r < ∞. Consider the Volterra operator
Vr : Lr [0, 1] → Lr [0, 1] given by Vr (f )(x) := x
0 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 measure
νr : 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.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 5/14

16. Theorem: Let 1 ≤ p < ∞. Let T : X → (E, τ) be a continuous operator and let U be
a basis of absolutely convex open neighborhoods of 0 in (E, τ). Consider the following
assertions:
(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(gU BX[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(KU BX[1/p]
) + U.
Then (a) implies (b) and (c). If besides τ is metrizable then (b) or (c) implies (a).
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 6/14

17. Lemma: Let 1 < p < ∞. Let X(µ) be an order continuous Banach function space
and E be a Banach space. A continuous operator T : X(µ) → E is essentially
compact if and only if for every h ∈ X[1/p ] the map Th : X[1/p] → E given by
Th(·) := T(h ·), is compact.
Corollary: Let 1 < p < ∞. Let X(µ) be an order continuous Banach function space
and E be a Banach space. If T : X(µ) → E is a continuous essentially compact
operator then the restriction T|X[1/p]
: X[1/p] → E is compact.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 7/14

18. Lemma: Let 1 < p < ∞. Let X(µ) be an order continuous Banach function space
and E be a Banach space. A continuous operator T : X(µ) → E is essentially
compact if and only if for every h ∈ X[1/p ] the map Th : X[1/p] → E given by
Th(·) := T(h ·), is compact.
Corollary: Let 1 < p < ∞. Let X(µ) be an order continuous Banach function space
and E be a Banach space. If T : X(µ) → E is a continuous essentially compact
operator then the restriction T|X[1/p]
: X[1/p] → E is compact.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 7/14

19. Theorem: Let 1 ≤ p < ∞ and let X(µ) be an order continuous Banach function
space. 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 that
T(BX ) ⊂ T(hεBX[1/p]
) + εBE .
(iii) T is essentially compact and for every ε > 0 there exists Kε > 0 such that
T(BX ) ⊂ T(KεBX[1/p]
) + εBE .
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 8/14

20. KERNEL OPERATORS
Corollary:Let X(µ), F and T as above. Suppose also that T is essentially compact.
Then
lim
K→∞
sup
f ∈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 operator
T : X(µ) → Y (ν) such that the kernel k satisfies that
lim
µ(A)→0
χAk(x, y) X (y)
Y
= 0.
Then T is compact.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 9/14

21. KERNEL OPERATORS
Corollary:Let X(µ), F and T as above. Suppose also that T is essentially compact.
Then
lim
K→∞
sup
f ∈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 operator
T : X(µ) → Y (ν) such that the kernel k satisfies that
lim
µ(A)→0
χAk(x, y) X (y)
Y
= 0.
Then T is compact.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 9/14

22. KERNEL OPERATORS
Corollary:Let X(µ), F and T as above. Suppose also that T is essentially compact.
Then
lim
K→∞
sup
f ∈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 operator
T : X(µ) → Y (ν) such that the kernel k satisfies that
lim
µ(A)→0
χAk(x, y) X (y)
Y
= 0.
Then T is compact.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 9/14

23. KERNEL OPERATORS
Corollary:Let X(µ), F and T as above. Suppose also that T is essentially compact.
Then
lim
K→∞
sup
f ∈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 operator
T : X(µ) → Y (ν) such that the kernel k satisfies that
lim
µ(A)→0
χAk(x, y) X (y)
Y
= 0.
Then T is compact.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 9/14

24. COMPACT OPTIMAL DOMAINS FOR ESSENTIALLY COMPACT OPERATORS
Characterize when the optimal domain of an operator is an L1-space. Consider a
vector measure m : Σ → E and let Im : L1(m) → E be the integration operator
Im(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 that
Im(BL1(m)) ⊂ Im(KεBLp(m)) + εBE .
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 10/14

25. COMPACT OPTIMAL DOMAINS FOR ESSENTIALLY COMPACT OPERATORS
Characterize when the optimal domain of an operator is an L1-space. Consider a
vector measure m : Σ → E and let Im : L1(m) → E be the integration operator
Im(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 that
Im(BL1(m)) ⊂ Im(KεBLp(m)) + εBE .
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 10/14

26. COMPACT OPTIMAL DOMAINS FOR ESSENTIALLY COMPACT OPERATORS
Characterize when the optimal domain of an operator is an L1-space. Consider a
vector measure m : Σ → E and let Im : L1(m) → E be the integration operator
Im(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 that
Im(BL1(m)) ⊂ Im(KεBLp(m)) + εBE .
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 10/14

27. Corollary:Let m : Σ → E be a vector measure with relatively compact range. Suppose
that 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?
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 11/14

28. Corollary:Let m : Σ → E be a vector measure with relatively compact range. Suppose
that 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?
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 11/14

29. Theorem: Let 1 < p < ∞, let X(µ) be an order continuous Banach function space, E
a Banach space and T : X(µ) → E a continuous operator. The following statements
are equivalent for T.
(i) T is essentially compact, and for each ε > 0 there is hε ∈ X[1/p ] such that
sup
f ∈BL1(mT )
∩X(µ)
inf
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
sup
f ∈BL1(mT )
∩X(µ)
inf
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. T
factorizes compactly as
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 12/14

30. Corollary: Let 1 < p < ∞, let X(µ) be an order continuous Banach function space, E
a Banach space and T : X(µ) → E a µ-determined essentially compact continuous
operator. The following statements are equivalent.
(i) Each extension of T to an order continuous Banach function space satisfies a
1/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.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 13/14

31. J. M. Calabuig, O. Delgado and E. A. S´anchez P´erez, Factorizing operators on
Banach function spaces through spaces of multiplication operators, J. Math.
Anal. Appl. 364 (2010), 88-103.
I. Ferrando and J. Rodr´ıguez, 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 compact
linear extension?, in: Operator Theory: Advances and Applications, 201.
Birkh¨auser Verlag, Basel, 2009, 313–322.
A. R. Schep, When is the optimal domain of a positive linear operator a weighted
L1-space?, in: Operator Theory: Advances and Applications, 201. Birkh¨auser
Verlag, Basel, 2009, 361-369.
E. S´anchez — p-potencias y compacidad para operadores entre espacios de Banach de funciones 14/14