Geometric properties of Banach function spaces of vector measure p-integrable functions

1. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Geometric properties of Banach function spaces of vector measure p-integrable functions Enrique A. Sánchez Pérez Instituto Universitario de Matemática Pura y Aplicada (I.U.M.P.A.), Universidad Politécnica de Valencia. Bangalore 2010 Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

2. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Let (Ω,Σ) be a measurable space, X a Banach space and m : Σ → X a vector measure. Let 1 ≤ p < ∞, and consider the space Lp(m) of p-integrable functions with respect to m. It is well known that these spaces are order continuous p-convex Banach function spaces with respect to µ, where µ is a Rybakov measure for m. In fact, each Banach lattice having these properties can be written (order isomorphically) as an Lp(m) space for a positive vector measure m. In this talk we explain further geometric properties of these spaces, their subspaces and the seminorms that define the weak topology on Lp(m). The main applications that are shown are related to the factorization and extension theory of operators in these spaces. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

3. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Each p-convex Banach function space can be represented as a Lp(m), a space of p-integrable functions with respect to a vector measure m. This is the starting point of a representation theory for operators acting on spaces of integrable functions that allows the extension on the results that are known on factorization of operators defined between these spaces. Moreover, it provides a framework in which new problems on the structure of these function spaces appear. Consider a finite measure space (Ω,Σ,µ) and an order continuous Banach function space with a weak unit X(µ) over it. If T : X(µ) → E is an operator with values on a Banach space E, it is always possible to associate to T a vector measure mT ; if moreover the operator has the right properties, it can be factorized through Lp(mT ). Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

4. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Each p-convex Banach function space can be represented as a Lp(m), a space of p-integrable functions with respect to a vector measure m. This is the starting point of a representation theory for operators acting on spaces of integrable functions that allows the extension on the results that are known on factorization of operators defined between these spaces. Moreover, it provides a framework in which new problems on the structure of these function spaces appear. Consider a finite measure space (Ω,Σ,µ) and an order continuous Banach function space with a weak unit X(µ) over it. If T : X(µ) → E is an operator with values on a Banach space E, it is always possible to associate to T a vector measure mT ; if moreover the operator has the right properties, it can be factorized through Lp(mT ). Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

5. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators 1 If the operator is p-concave and X(µ) is a p-convex Banach function space, it is possible to obtain a strong factorization (defined by a multiplication operator) through the space Lp(µ). (Maurey-Rosenthal Theorems). A. Defant. 2 General framework: inequalities for operators and geometry. 3 Rosenthal, Krivine and Maurey (1970’s), Banach lattice properties, p-convexity, p-concavity. 4 Applications: almost everywhere convergence of sequences, unconditional convergence of sequences; characterizations of operators belonging to particular operator ideals... Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

9. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Index 1 Lp(m)-spaces and factorizations 2 Volterra operators 3 Applications: Strongly p-th power factorable operators Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

12. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Let (Ω,Σ,µ) be a measure space. X(µ) is a Banach function space over µ if: 1) If |f(ω)| ≤ |g(ω)| µ-a.e., where f ∈ L0(µ) and g ∈ X(µ), then f ∈ X(µ) and f ≤ g . 2) For each A ∈ Σ of finite measure, χA ∈ X(µ). If X(µ) is σ-order continuous (for each fn ↓ 0, l´ımn fn = 0), the dual of X(µ) equals its Köthe dual; each continuous functional ϕ ∈ X(µ)∗ an be written as an integral ϕ(f) = Ω fg dµ. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

13. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Let (Ω,Σ,µ) be a measure space. X(µ) is a Banach function space over µ if: 1) If |f(ω)| ≤ |g(ω)| µ-a.e., where f ∈ L0(µ) and g ∈ X(µ), then f ∈ X(µ) and f ≤ g . 2) For each A ∈ Σ of finite measure, χA ∈ X(µ). If X(µ) is σ-order continuous (for each fn ↓ 0, l´ımn fn = 0), the dual of X(µ) equals its Köthe dual; each continuous functional ϕ ∈ X(µ)∗ an be written as an integral ϕ(f) = Ω fg dµ. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

14. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators A Banach lattice E is p-convex if there is a constant K such that for each finite set x1,...,xn ∈ E, ( n ∑ i=1 |xi |p )1/p ≤ K( n ∑ i=1 xi p )1/p . Let T : E → F be an operator, where F is a lattice. T is p-concave if there is a constant K such that for each finite family x1,...,xn ∈ E, ( n ∑ i=1 T(xi ) p )1/p ≤ K ( n ∑ i=1 |xi |p )1/p . Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

15. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators A Banach lattice E is p-convex if there is a constant K such that for each finite set x1,...,xn ∈ E, ( n ∑ i=1 |xi |p )1/p ≤ K( n ∑ i=1 xi p )1/p . Let T : E → F be an operator, where F is a lattice. T is p-concave if there is a constant K such that for each finite family x1,...,xn ∈ E, ( n ∑ i=1 T(xi ) p )1/p ≤ K ( n ∑ i=1 |xi |p )1/p . Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

16. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Let X be a Banach space. We consider a countably additive vector measure m : Σ → X; i.e. if {Ai }∞ i=1 is a family of disjoint sets Σ, then m(∪∞ i=1Ai ) = ∞ ∑ i=1 m(Ai ). If m is a vector measure, we denote by |m| its variation and by m its semivariation. If x ∈ X∗, then m,x (A) := m(A),x , A ∈ Σ, defines a measure. A Rybakov measure for m is a measure as | m,x | that controls m. If X(µ) is an order continuous Banach function space and T : X(µ) → E is an operator, the expression mT (A) := T(χA), A ∈ Σ, defines a countably additive vector measure. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

19. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators A Σ-measurable function f is integrable with respect to a vector measure m : Σ → E if 1) f is m,x -integrable for each x ∈ X∗, and 2) for each A ∈ Σ there exists a unique element mf (A) ∈ X such that mf (A),x = A f d m,x , x ∈ X∗ . This element is usually denoted by A f dm. Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

22. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators The space L1(m) of all (classes of a.e. equal) of integrable functions with respect to a vector measure with the norm f L1(m) := sup x ∈BX∗ |f|d| m,x |, f ∈ L1 (m), is an o.c. Banach function space with weak unit over any Rybakov measure | m,x | de m. The space L1 w (m) is defined as the space of all (classes of) functions satisfying 1), with the same norm. If 1 < p < ∞, the space Lp(m) is defined in the same way; in this case, the norm is given by the expression f Lp(m) := sup x ∈BX ( Ω |f|p d| m,x |)1/p , f ∈ Lp (m). (1) Lp(m) is a p-convex o.c. Banach function space with weak unit over each Rybakov measure for m. L p w (m) is defined as the set of (classes of) measurable functions f such that |f|p is scalarly integrable; the norm is also given by (1). Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

25. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators If X(µ) a Banach function space we define its p-th power X(µ)[p] as the space X(µ)[p] := {|f|p : f ∈ X(µ)}. If X(µ) is p-convex, X(µ)[p] is a Banach function space over µ with the quasi norm g X(µ)[p] := |g|1/p p X(µ) , g ∈ X(µ)[p], that in this case is equivalent to a norm. If µ is a finite measure, then the inclusion X(µ) ⊆ X(µ)[p] is well-defined and continuous. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

28. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Theorem Let 1 < p < ∞. If E is a p-convex o. c. Banach lattice with weak unit, then there exists a positive vector measure m with values in E such that Lp(m) and E are lattice isomorphic. Demostración. If E is p-convex, it is always possible to find an equivalent lattice norm of E satisfying that its p-convexity constant is 1. There is a probability space (Ω,Σ,µ) such that E is lattice isomorphic and isometric to a Banach function space X(µ) (in particular, X(µ) is p-convex). The p-power X(µ)[p] of X(µ) is an order continuous Banach function space. The set function ν : Σ → X(µ)[p] is a vector measure such that L1(ν) = X(µ)[p]. So, X(µ) = X(µ)[p] [1/p] = L1(ν)[1/p] = Lp(ν). Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

32. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Operators factorizing through the p-th power Let µ be a finite measure. Let X(µ) be an o.c. Banach function space and E a Banach space. If 1 ≤ p < ∞, an operator T : X(µ) → E is p-th power factorable if T[p] : X(µ)[p] → E satisfies T = T[p] ◦ i[p], (2) where i[p] : X(µ) → X(µ)[p] denotes the inclusion map. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

33. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Operators factorizing through the p-th power Let µ be a finite measure. Let X(µ) be an o.c. Banach function space and E a Banach space. If 1 ≤ p < ∞, an operator T : X(µ) → E is p-th power factorable if T[p] : X(µ)[p] → E satisfies T = T[p] ◦ i[p], (2) where i[p] : X(µ) → X(µ)[p] denotes the inclusion map. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

34. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators i.e., the following diagram commutes X(µ) T E E i[p] r rrr rrj X(µ)[p] ¨¨ ¨¨¨¨B T[p] We denote by F[p](X(µ),E) the class of all p-th power factorable operators. We consider µ-determined operators: If mT (A) = 0 then µ(A) = 0 for each A ∈ Σ. Example: if µ is a finite measure and 1 < p < ∞, then the inclusion map i : Lp(µ) → L1(µ) is p-th power factorable. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

35. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators i.e., the following diagram commutes X(µ) T E E i[p] r rrr rrj X(µ)[p] ¨¨ ¨¨¨¨B T[p] We denote by F[p](X(µ),E) the class of all p-th power factorable operators. We consider µ-determined operators: If mT (A) = 0 then µ(A) = 0 for each A ∈ Σ. Example: if µ is a finite measure and 1 < p < ∞, then the inclusion map i : Lp(µ) → L1(µ) is p-th power factorable. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

36. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Theorem Let 1 ≤ p < ∞, and let X(µ) be an o.c. Banach function space over a finite measure µ and E a Banach space. Let T : X(µ) → E be µ-determined. Then the following are equivalent. (i) T factorizes through the p-th power. (ii) There is a constant C > 0 such that T(f) E ≤ C f X(µ)[p] = C |f|1/p p X(µ) , f ∈ X(µ). (3) (iii) X(µ) ⊆ Lp(mT ), and the inclusion map is continuous. (iv) T factorizes as X(µ) T E E i rr rrrrj Lp(mT ) ¨ ¨¨¨ ¨¨B I (p) mT Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

41. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators (v) X(µ)[p] ⊆ L1(mT ), and the inclusion is continuous. (vi) X(µ) ⊆ L p w (mT ) and the inclusion is continuous. (vii) X(µ)[p] ⊆ L1 w (mT ) and the inclusion is continuous. (viii) For each x ∈ E∗, the Radon-Nikodým derivative d mT ,x dµ belongs to the Köthe dual X(µ)[p] of X(µ)[p]. Remark: the p-th power is not necessarily a Banach function space; in general it is only a quasi-normed lattice. For instance L1(µ)[p] = L1/p(µ). Duality? Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

42. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators (v) X(µ)[p] ⊆ L1(mT ), and the inclusion is continuous. (vi) X(µ) ⊆ L p w (mT ) and the inclusion is continuous. (vii) X(µ)[p] ⊆ L1 w (mT ) and the inclusion is continuous. (viii) For each x ∈ E∗, the Radon-Nikodým derivative d mT ,x dµ belongs to the Köthe dual X(µ)[p] of X(µ)[p]. Remark: the p-th power is not necessarily a Banach function space; in general it is only a quasi-normed lattice. For instance L1(µ)[p] = L1/p(µ). Duality? Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

43. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Optimal Domain Theorem. Theorem Let 1 ≤ p < ∞ and T ∈ L X(µ),E a µ-determined p-th power factorable operator. Then Lp(mT ) is the biggest o.c. Banach function space with weak unit over µ to which one T can be extended, with the extension still satisfying that is p-th power factorable. Demostración. Suppose that Y(µ) is a Banach function space such that X(µ) ⊆ Y(µ), T can be extended to it and the extension is still p-th power factorable. Let T ∈ L Y(µ),E be such an extension. Then mT = mT . In particular, T is also µ-determined. Thus, by the characterization theorem for operators factorizing through the p-th power, Y(µ) ⊆ Lp(mT ) = Lp(mT ). This proves the theorem Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

48. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Example: Volterra operators. Are Volterra operators p-th power factorable? Let µ be Lebesgue measure on Ω := [0,1] Σ := B([0,1]) be the Borel σ-algebra. For 1 ≤ r < ∞, consider the Volterra operator Vr : Lr ([0,1]) → Lr ([0,1]) given by Vr (f)(t) := t 0 f(u)du, t ∈ [0,1], f ∈ Lr ([0,1]). Vr is µ-determined. The associated vector measure mVr : A → Vr (χA) in Σ coincides with the Volterra measure νr of order r, i.e. mVr (A) = Vr (χA) = νr (A), A ∈ Σ; (4) The variation |νr | of νr is finite and is given by d|νr |(t) = (1−t)1/r dt. So, Lr ([0,1]) ⊆ L1 ([0,1]) ⊆ L1 ((1−t)1/r dt) ⊆ L1 (νr ). (5) Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

53. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators It can be proved that Lq([0,1]) ⊆ L1([0,1]) ⊆ L1((1−t)dt) whenever 1 ≤ q ≤ ∞, and moreover Lq ([0,1]) L1 ((1−t)dt) whenever 0 < q < 1, (6) which can be deduced from the fact that t → t−1 · χ[0,1](t) belongs to Lq([0,1]) but not to L1([0,1]). (i) Let r := 1. If X(µ) := L1([0,1]) we have that L1(|ν1|) = L1(ν1) and so for each 1 < p < ∞, X(µ)[p] = L1/p ([0,1]) L1 (1−t)dt = L1 (ν1) = L1 (mV1 ) Then the characterization theorem V1 is not p-th power factorable. Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

54. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators It can be proved that Lq([0,1]) ⊆ L1([0,1]) ⊆ L1((1−t)dt) whenever 1 ≤ q ≤ ∞, and moreover Lq ([0,1]) L1 ((1−t)dt) whenever 0 < q < 1, (6) which can be deduced from the fact that t → t−1 · χ[0,1](t) belongs to Lq([0,1]) but not to L1([0,1]). (i) Let r := 1. If X(µ) := L1([0,1]) we have that L1(|ν1|) = L1(ν1) and so for each 1 < p < ∞, X(µ)[p] = L1/p ([0,1]) L1 (1−t)dt = L1 (ν1) = L1 (mV1 ) Then the characterization theorem V1 is not p-th power factorable. Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

55. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators (ii) Let 1 < r < ∞. For 1 < p < ∞, it can be proved that Vr is p-th power factorable if and only if p ≤ r. (a) Suposse that Vr is p-th power factorable. Let S : Lr ([0,1]) → L1 ([0,1]) be the inclusion map. Then S ◦Vr : Lr ([0,1]) → L1 ([0,1]) is p-th power factorable The associated vector measure mS◦Vr de S ◦Vr equals ν1. Then, the characterization theorem gives Lr/p ([0,1]) = X(µ)[p] ⊆ L1 (mS◦Vr ) = L1 (ν1) = L1 (1−t)dt). but only in the case that r/p ≥ 1, i.e. p ≤ r. (b) Suppose now that p ≤ r. Then for X(µ) := Lr ([0,1]) we have X(µ)[p] = Lr/p ([0,1]) ⊆ L1 ([0,1]) ⊆ L1 (νr ) = L1 (mVr ). So by the characterization theorem Vr is p-th power factorable. Lp(mT ) is not the biggest p-convex o.c. B.f.s for extending T. Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

59. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators A Maurey-Rosenthal type theorem Definition Let 0 < q < ∞ and 1 ≤ p < ∞. We say that a continuous linear operator T : X(µ) → E from a q-B.f.s. X(µ) into a Banach space E is bidual (p,q)-power-concave if there exists a constant C1 > 0 such that n ∑ j=1 T(fj ) q/p E ≤ C1 n ∑ j=1 fj q/p b,X(µ)[q] , f1,...,fn ∈ X(µ), n ∈ N, (7) where · b,X(µ)[q] denotes the norm in the bidual of X(µ)[q]. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

60. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators A Maurey-Rosenthal type theorem Definition Let 0 < q < ∞ and 1 ≤ p < ∞. We say that a continuous linear operator T : X(µ) → E from a q-B.f.s. X(µ) into a Banach space E is bidual (p,q)-power-concave if there exists a constant C1 > 0 such that n ∑ j=1 T(fj ) q/p E ≤ C1 n ∑ j=1 fj q/p b,X(µ)[q] , f1,...,fn ∈ X(µ), n ∈ N, (7) where · b,X(µ)[q] denotes the norm in the bidual of X(µ)[q]. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

61. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Example of bidual (p,q)-power-concave operator. Let 1 ≤ p < q and consider two finite measure spaces (Ω1,Σ1,µ1) and (Ω2,Σ2,µ2). Take a Bochner integrable function φ ∈ Lq/p(µ2,Lq (µ1)). Define the operator uφ : Lpq(µ1) → Lq/p(µ2) by uφ (f)(w2) := f,φ(w2) = Ω1 f(w1)(φ(w2)(w1))dµ1(w1) ∈ Lq/p (µ2), f ∈ Lpq(µ1). The operator uφ is well-defined and continuous since Lpq(µ1) ⊆ Lq(µ1). Consider a finite set of functions f1,...,fn ∈ Lpq(µ1). Then n ∑ i=1 uφ (fi ) q/p Lq/p(µ2) p/q ≤ Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

62. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators = Ω2 n ∑ i=1 | fi , φ(w2) φ(w2) |q/p φ(w2) q/p dµ2(w2) p/q ≤ sup h Lq (µ1) ≤1 n ∑ i=1 | fi ,h |q/p p/q · φ Lq/p(µ2,Lq (µ1)) ≤ n ∑ i=1 fi q/p p/q Lq(µ1) · φ = n ∑ i=1 fi q/p p/q Lpq(µ1)[q] · φ Lq/p(µ2,Lq (µ1)) , Therefore, uφ is (p,q)-power concave, and since Lpq(µ1) is q-convex, it is also bidual (p,q)-power concave. Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

63. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators In the case the Bochner integrable function is given by a kernel, i.e. φ(w2)(w1) := K (w1,w2), K being a µ1 × µ2 measurable function satisfying |K (w1,w2)|q dµ1(w1) q/(pq ) dµ2(w2) p/q < ∞, then uφ is a so called Hille-Tamarkin operator. For the case p = 1 and µ1 = µ2, the class of Hille-Tamarkin operators is a well-known class of classical kernel operators that has been largely studied. Volterra operators are particular cases of this class. Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

64. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators In the case the Bochner integrable function is given by a kernel, i.e. φ(w2)(w1) := K (w1,w2), K being a µ1 × µ2 measurable function satisfying |K (w1,w2)|q dµ1(w1) q/(pq ) dµ2(w2) p/q < ∞, then uφ is a so called Hille-Tamarkin operator. For the case p = 1 and µ1 = µ2, the class of Hille-Tamarkin operators is a well-known class of classical kernel operators that has been largely studied. Volterra operators are particular cases of this class. Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

65. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Theorem Let X(µ) be a σ-order continuous q-B.f.s. over a finite measure space (Ω,Σ,µ). Let E be a Banach space and an operator T ∈ L (X(µ),E) be µ-determined. For any 1 ≤ p < ∞ and 0 < q < ∞, the following assertions are equivalent. (i) There exists C > 0 such that n ∑ j=1 T(fj ) q/p E 1/q ≤ C n ∑ j=1 fj q/p 1/q b,X(µ)[q] . for all n ∈ N and f1,...,fn ∈ X(µ); namely, T is bidual (p,q)-power-concave. (ii) There exists a function g ∈ X(µ)[q] with g > 0 (µ-a.e.) such that T(f) E ≤ Ω |f|q/p g dµ p/q < ∞, f ∈ X(µ). (8) Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

66. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators (iii) The inclusions X(µ) ⊆ Lq(g dµ) ⊆ Lp(mT ) hold and are continuous for some g ∈ L0(µ) with g > 0 (µ-a.e.). (iv) The inclusions X(µ)[p] ⊆ Lq/p(gdµ) ⊆ L1(mT ) hold and are continuous for some g ∈ L0(µ) with g > 0 (µ-a.e.). (v) T ∈ F[p](X(µ),E) and there exist an operator S ∈ L (Lq/p(µ),E) and a function g > 0 (µ-a.e.) satisfying gp/q ∈ M X(µ)[p], Lq/p(µ) such that T[p] = S ◦Mgp/q . That is, the following diagram commutes: X(µ)[p] X(µ) i[p] c S E Mgp/q Lq/p(µ), EET T ¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨B T[p] Enrique A. S´anchez P´erez Geometric properties of Banach function spaces of vector measure p-inte

67. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators (vi) T ∈ F[p] X(µ),E and the inclusion map J (p) T : X(µ) → Lp(mT ) is bidual q-concave. (vii) T ∈ F[p] X(µ),E and the inclusion map β[p] : X(µ)[p] → L1(mT ) is bidual (q/p)-concave. (viii) T ∈ F[p] X(µ),E and T[p] : X(µ)[p] → E is bidual (q/p)-concave. If T ∈ L (X(µ),E) satisfies any one of (i)–(viii), then the following diagram commutes: X(µ)[p] X(µ) c E ELq/p(gdµ) E Lq(gdµ) E L1(mT ) E E Lp(mT ) c with each arrow indicating the respective inclusion map. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

68. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Definition A family of R-valued functions Ψ is called concave if, for every finite set of functions {ψ1,...,ψn} ⊆ Ψ with n ∈ N and non-negative scalars c1,...,cn satisfying ∑n j=1 cj = 1, there exists ψ ∈ Ψ such that n ∑ j=1 cj ψj ≤ ψ. The following result is known as Ky Fan’s Lemma. Lemma Let W be a compact convex subset of a Hausdorff topological vector space and let Ψ be a concave collection of lower semi-continuous, convex, real functions on W. Let c ∈ R. Suppose, for every ψ ∈ Ψ, that there exists xψ ∈ W with ψ(xψ ) ≤ c. Then there exists x ∈ W such that ψ(x) ≤ c for all ψ ∈ Ψ. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

69. Lp(m)-spaces and factorizations Volterra operators Applications: Strongly p-th power factorable operators Some references A. Defant and E.A. Sanchez Perez, Domination of operators on Banach function spaces. Math. Proc. Cambridge P. Soc., 146, 57-66(2009). S. Okada, W. J. Ricker and E. A. Sánchez-Pérez, Optimal domain and integral extension of operators —Acting in function spaces—, Operator Theory: Advances and Applications, vol. 180, Birkhäuser Verlag, Basel, 2008. Enrique A. Sánchez Pérez Geometric properties of Banach function spaces of vector measure p-inte

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