Dominación y extensiones óptimas de operadores con rango esencial compacto en espacios de Banach de funciones

1. Dominación y extensiones óptimas de operadores con rango esencial compacto en espacios de Banach de funciones P. Rueda, Enrique A. Sánchez Pérez Seminario de Medidas Vectoriales en Integración I.U.M.P.A.-U. Politécnica de Valencia Diciembre 2012 E. Sánchez Extensiones óptimas de operadores

2. Let 1 < p,p < ∞ such that 1/p +1/p = 1. We say that an operator T from X(µ) into a Banach space E satisfies a 1/p-th power approximation if for each ε > 0 there is a function hε ∈ X(µ)[1/p ] such that for every function f ∈ BX(µ) there is a function g ∈ BX(µ)[1/p] such that T(f −ghε ) E < ε. problem: Let X(µ) be an order continuous Banach function space over a finite measure µ and let T : X(µ) → E be a Banach space valued operator with compact essential range. When is the maximal extension of T compact? E. Sánchez Extensiones óptimas de operadores

3. Let 1 < p,p < ∞ such that 1/p +1/p = 1. We say that an operator T from X(µ) into a Banach space E satisfies a 1/p-th power approximation if for each ε > 0 there is a function hε ∈ X(µ)[1/p ] such that for every function f ∈ BX(µ) there is a function g ∈ BX(µ)[1/p] such that T(f −ghε ) E < ε. problem: Let X(µ) be an order continuous Banach function space over a finite measure µ and let T : X(µ) → E be a Banach space valued operator with compact essential range. When is the maximal extension of T compact? E. Sánchez Extensiones óptimas de operadores

4. Motivation: Compactness. Grothendieck’s result. Theorem Let E be a Banach space and let K ⊆ E be a weakly closed subset. If for every ε > 0 there exists a weakly compact subset K(ε) of X such that K ⊆ K(ε)+εBE , then K is weakly compact. E. S´anchez Extensiones ´optimas de operadores

5. Theorem Let 1 ≤ p < ∞ and X an order continuous Banach function space over a finite measure space. 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. Sánchez Extensiones óptimas de operadores

6. Proof. (a)⇒(b). If T : X → (E,τ) is compact then T(gBX[1/p] ) is relatively τ-compact in E for all g ∈ BX[1/p ] . Take U ∈ U . Then T(BX ) τ ⊂ T(BX )+U ⊂ T(∪g∈BX[1/p ] gBX[1/p] )+U = ∪g∈BX[1/p ] (T(gBX[1/p] )+U). By (a) T(BX ) τ is τ-compact. Since each set of the form T(gBX[1/p] )+U is τ-open, there exist finitely many g1,...,gl ∈ BX[1/p ] such that T(BX ) τ ⊂ ∪l i=1T(gi BX[1/p] )+U. By the lattice properties of the norm of X[1/p], for any h1,h2 ∈ X[1/p ] satisfying |h1| ≤ |h2| a.e., we have that h1BX[1/p] ⊆ |h2|BX[1/p] . Define gU = ∑l i=1 |gi |. Since |gi | ≤ gU , we obtain that gi BX[1/p] ⊂ gU BX[1/p] , and so T(gi BX[1/p] ) ⊂ T(gU BX[1/p] ) for all i = 1,...,l. Therefore, ∪l i=1T(gi BX[1/p] )+U ⊂ T(gU BX[1/p] )+U and (b) is proved. E. Sánchez Extensiones óptimas de operadores

12. (b)⇒(c) Let ε > 0 and U ∈ U . Take V ∈ U such that V +V ⊂ U. Consider gV ∈ X[1/p ] given by (b). By continuity there exists δ > 0 such that T(δBX ) ⊂ V. Since simple functions are dense in X[1/p ], we find a simple function s such that (gV −s)f X ≤ gV −s X[1/p ] < δ for every f ∈ BX[1/p] , and so T((gV −s)BX[1/p] ) ⊆ V. Take KU := s L∞(µ). Then, again by the ideal property of X[1/p], we obtain T(BX ) ⊆ T(gV BX[1/p] )+V ⊆ T(sBX[1/p] )+T((gV −s)BX[1/p] )+V ⊆ T(sBX[1/p] )+V +V ⊆ T(KU BX[1/p] )+U. The proof is finished Let us assume now that τ is metrizable. We prove (b)⇒(a) (the proof for (c)⇒(a) is the same). Take an arbitrary U ∈ U and chose gU ∈ X[1/p ] such that T(BX ) ⊂ T(gU BX[1/p] )+U. By (b) the set T(gU BX[1/p] ) is relatively τ-compact in E. Then we have shown that for any U ∈ U , there is a relatively τ-compact set T(gU BX[1/p] ) such that T(BX ) ⊂ T(gU BX[1/p] )+U. As τ is metrizable this implies that T(BX ) is relatively τ-compact. E. Sánchez Extensiones óptimas de operadores

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 Extensiones ´optimas de operadores

19. 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 Extensiones ´optimas de operadores

20. 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(hε BX[1/p] )+εBE . (iii) T is essentially compact and for every ε > 0 there exists Kε > 0 such that T(BX ) ⊂ T(Kε BX[1/p] )+εBE . E. S´anchez Extensiones ´optimas de operadores

21. Examples (a) Take X(µ) = L1[0,1] and 1 < p < ∞. Then X[1/p] = Lp[0,1] isometrically. Consider a partition {Ai }∞ i=1 of [0,1], where µ(Ai ) > 0 for all i ∈ N. Take a sequence of non-null measurable sets {Bi }∞ i=1 such that Bi ⊆ Ai for all i and ri := µ(Bi )/µ(Ai ) ↓ 0 and the operator T : L1(µ) → 1 given by T(f) := ∑∞ i=1( Bi f dµ)ei ∈ 1, f ∈ L1[0,1]. Let us show that the requirement on {T(χA) : A ∈ Σ} of being relatively compact is fulfilled. Consider the sequence {ri ei }∞ i=1. For every A ∈ Σ, T(χA) = ∞ ∑ i=1 µ(Bi ∩A)ei ≤ ∞ ∑ i=1 µ(Ai )(ri ei ) in the order of 1, and so each T(χA) is in the closure of the convex hull of {ri ei }∞ i=1. Thus, it is relatively compact. In fact, any 1-valued continuous operator defined on an order continuous Banach function space is essentially compact. However, since for every i we can find a norm one function fi of L1(µ) with support in Bi , the set T(BL1[0,1]) includes all {ei }∞ i=1, and so T is not compact. Consequently, there is ε > 0 such that there is no K such that T(BL1[0,1]) ⊆ KT(BLp[0,1])+εB p . E. Sánchez Extensiones óptimas de operadores

22. (b) Although operators of compact range with domain in an L1-space allow sometimes good characterizations —due mainly to 1-concavity of L1—, this fact is not crucial in the example above. Consider p ≥ q ≥ 1 and the class of spaces Ep,q = (⊕p)∞ i=1Lq(Ai ,µ|Ai ), where the Ai s are chosen as in (a), with the norm | f |p,q := ∞ ∑ i=1 f|Ai p Lq[0,1] 1/p , f ∈ Ep,q. It is a Banach function space of Lebesgue measurable functions on [0,1] over a measure µa given by µa(A) := ∑∞ i=1 ai µ(Ai ∩A), where a is any strictly positive element of B p . Notice also that it contains L1[0,1]. The same calculations that in the above case prove that T defined as in (a) is continuous also if defined from Ep,1 to p, and its essential range T is relatively compact, but the operator is not compact. Then, since (Ep,1)[1/p] = Ep2,p, we get that there is an ε > 0 such that T(BEp,1 ) KT(BEp2,p )+εB 1 for all constants K > 0. E. Sánchez Extensiones óptimas de operadores

23. Corollary Let 1 < p < ∞. Let X(µ), E and T as above, and assume that T is essentially compact. Let Φ : BX → BX[1/p] be a function and suppose that l´ım K→∞ sup f∈BX T(f χ{|f|≥K|Φ(f)|}) = 0. Then T is compact. Corollary Let T be an essentially compact (positive) kernel operator T : X(µ) → Y(ν) such that the kernel k satisfies that l´ım µ(A)→0 χAk(x,y) X (y) Y = 0. Then T is compact. E. Sánchez Extensiones óptimas de operadores

24. Corollary Let 1 < p < ∞. Let X(µ), E and T as above, and assume that T is essentially compact. Let Φ : BX → BX[1/p] be a function and suppose that l´ım K→∞ sup f∈BX T(f χ{|f|≥K|Φ(f)|}) = 0. Then T is compact. Corollary Let T be an essentially compact (positive) kernel operator T : X(µ) → Y(ν) such that the kernel k satisfies that l´ım µ(A)→0 χAk(x,y) X (y) Y = 0. Then T is compact. E. Sánchez Extensiones óptimas de operadores

25. 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(gε BLp(m))+εBE . (c) R(m) is relatively compact and for every ε > 0 there exists Kε > 0 such that Im(BL1(m)) ⊂ Im(Kε BLp(m))+εBE . E. S´anchez Extensiones ´optimas de operadores

26. Let 1 < p,p < ∞ such that 1/p +1/p = 1. We say that an operator T from X(µ) into a Banach space E satisfies a 1/p-th power approximation if for each ε > 0 there is a function hε ∈ X(µ)[1/p ] such that for every function f ∈ BX(µ) there is a function g ∈ BX(µ)[1/p] such that T(f −ghε ) E < ε. 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|). E. Sánchez Extensiones óptimas de operadores

27. Let 1 < p,p < ∞ such that 1/p +1/p = 1. We say that an operator T from X(µ) into a Banach space E satisfies a 1/p-th power approximation if for each ε > 0 there is a function hε ∈ X(µ)[1/p ] such that for every function f ∈ BX(µ) there is a function g ∈ BX(µ)[1/p] such that T(f −ghε ) E < ε. 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|). E. Sánchez Extensiones óptimas de operadores

28. 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) ´ınf g∈BLp(mT )∩X[1/p] T(f −hε g) 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) ´ınf g∈BLp(mT )∩X[1/p] T(f −Kε g) E < ε. (iii) The optimal domain of T is L1(|mT |) and the extension ImT is compact, i.e. T factorizes compactly as X(µ) T // r [i] $$HHHHHHHHH E. L1(|mT |) ImT ;;xxxxxxxxx E. S´anchez Extensiones ´optimas de operadores

29. 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. Sánchez Extensiones óptimas de operadores

30. E. S´anchez Extensiones ´optimas de operadores

31. We show a factorization theorem for homogeneous maps. The proof is based in a separation argument that has shown to be the key for proving factorization theorems for operators between Banach function spaces. Proposition. Let E be a Banach space, and let Y(ν) and Z(ν) be Banach function spaces such that Y(ν) ⊆ Z(ν). Let U ⊆ E be an homogeneous set, and φ : U → Y(ν) and P : U → Z(ν) be bounded homogeneous maps. Assume also that YZ has the Fatou property and (YZ ) is order continuous. Then the following statements are equivalent. (i) There is a constant K 0 such that for every x1,...,xn ∈ U and A1,...,An ∈ Σ, n ∑ i=1 |P(xi )|χAi L1(ν) ≤ K n ∑ i=1 |φ(xi )|χAi (YZ ) . (ii) There is a function g ∈ YZ such that for every x ∈ U there is a function hx ∈ BL∞(ν) depending only on x/ x such that P(x) = g ·hx ·φ(x). In other words, P factorizes through the homogeneous map ˆφ given by ˆφ(x) := hx ·φ(x) as U P // p ˆφ !!BBBBBBBB Z(ν) Y(ν) g yyyyyyyy E. S´anchez Extensiones ´optimas de operadores

32. We show a factorization theorem for homogeneous maps. The proof is based in a separation argument that has shown to be the key for proving factorization theorems for operators between Banach function spaces. Proposition. Let E be a Banach space, and let Y(ν) and Z(ν) be Banach function spaces such that Y(ν) ⊆ Z(ν). Let U ⊆ E be an homogeneous set, and φ : U → Y(ν) and P : U → Z(ν) be bounded homogeneous maps. Assume also that YZ has the Fatou property and (YZ ) is order continuous. Then the following statements are equivalent. (i) There is a constant K 0 such that for every x1,...,xn ∈ U and A1,...,An ∈ Σ, n ∑ i=1 |P(xi )|χAi L1(ν) ≤ K n ∑ i=1 |φ(xi )|χAi (YZ ) . (ii) There is a function g ∈ YZ such that for every x ∈ U there is a function hx ∈ BL∞(ν) depending only on x/ x such that P(x) = g ·hx ·φ(x). In other words, P factorizes through the homogeneous map ˆφ given by ˆφ(x) := hx ·φ(x) as U P // p ˆφ !!BBBBBBBB Z(ν) Y(ν) g yyyyyyyy E. S´anchez Extensiones ´optimas de operadores

33. DEFINITIONS: Banach function spaces (Ω,Σ,µ) be a finite measure space. L0(µ) space of all (classes of) measurable real functions on Ω. A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) with norm · X such that if f ∈ L0(µ), g ∈ X and |f| ≤ |g| µ-a.e. then f ∈ X and f X ≤ g X . A B.f.s. X has the Fatou property if for every sequence (fn) ⊂ X such that 0 ≤ fn ↑ f µ-a.e. and supn fn X ∞, it follows that f ∈ X and fn X ↑ f X . We will say that X is order continuous if for every f,fn ∈ X such that 0 ≤ fn ↑ f µ-a.e., we have that fn → f in norm. E. Sánchez Extensiones óptimas de operadores

39. APPENDIX: DEFINITIONS. Vector measures and integration Let m : Σ → E be a vector measure, that is, a countably additive set function, where E is a real Banach space. A set A ∈ Σ is m-null if m(B) = 0 for every B ∈ Σ with B ⊂ A. For each x∗ in the topological dual E∗ of E, we denote by |x∗m| the variation of the real measure x∗m given by the composition of m with x∗. There exists x∗ 0 ∈ E∗ such that |x∗ 0 m| has the same null sets as m. We will call |x∗ 0 m| a Rybakov control measure for m. A measurable function f : Ω → R is integrable with respect to m if (i) |f|d|x∗ m| ∞ for all x∗ ∈ E∗ . (ii) For each A ∈ Σ, there exists xA ∈ E such that x∗ (xA) = A f dx∗ m, for all x∗ ∈ E. The element xA will be written as A f dm. E. S´anchez Extensiones ´optimas de operadores

44. DEFINITIONS: Spaces of integrable functions Denote by L1(m) the space of integrable functions with respect to m, where functions which are equal m-a.e. are identified. The space L1(m) is a Banach space endowed with the norm f m = sup x∗∈BE∗ |f|d|x∗ m|. Note that L∞(|x∗ 0 m|) ⊂ L1(m). In particular every measure of the type |x∗m| is finite as |x∗m|(Ω) ≤ x∗ · χΩ m. Given f ∈ L1(m), the set function mf : Σ → E given by mf (A) = A f dm for all A ∈ Σ is a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in this case g dmf = gf dm. E. Sánchez Extensiones óptimas de operadores

Dominación y extensiones óptimas de operadores con rango esencial compacto en espacios de Banach de funciones

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