Optimal Domain and Integral Extension of Operators

Okada, Susumu; Ricker, Werner J.; Pérez, Enrique Alfonso Sánchez

doi:10.1007/978-3-7643-8648-1

Cited by 165 publications

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“…The main references on vector measures used in this paper are [9,24]. Let m : Σ → E be a vector measure, where E is a Banach space.…”

Section: Preliminariesmentioning

confidence: 99%

“…m T and μ have the same null sets, in this case X(μ) ⊆ L 1 (m T ) continuously (see [24,Chapter 4]). This property can be removed, but in this case we will have a continuous map not necessarily injective (see [13,Lemma 3.2] and the comments above it and below it).…”

Section: Preliminariesmentioning

confidence: 99%

“…In Lemma 3.1 we have assumed the condition (3.1). This may happen, for instance when the vector measure is the associated with a so-called p-th power factorable operator defined on a suitable B.f.s., see [24,Theorem 5.7(vii)] for more information on this class of operators.…”

Section: Mjommentioning

confidence: 99%

“…This problem is the inverse of the classical one in spectral theory. What follows is a simple example for the case of a separable kernel operator (sometimes called degenerate or of finite rank), that provides us a set of possible vector measures that satisfies the conditions of the Theorem 4.1, whenever the operator has the μ-determinacy conditions (see [24,Proposition 4.23] for some conditions on this second aspect).…”

Section: Mjommentioning

confidence: 99%

“…'s such that X(μ) is o.c. and T k : X(μ) → Y (μ) be a μ-determined kernel operator, then T K admits the integral extension I mT K : L 1 (m T k ) → Y (μ) ( [24,Theorem 4.14]). Let h 0 ∈ Y (μ) be such that h 0 = I mT K (f ) for some unknown f ∈ L 1 (m T k ), thus we obtain the following equivalence for an integral equation…”

Section: The Integral Operator Equationmentioning

confidence: 99%

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On the Norm with Respect to Vector Measures of the Solution of an Infinite System of Ordinary Differential Equations

Galdames-Bravo

2014

Mediterr. J. Math.

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In the present paper we give some necessary conditions that satisfy the solutions of an infinite system of ordinary differential equations. We investigate the behavior of the solutions of a general system of equations, regarding the norm of a Banach function space based on a vector measure. To this aim we construct a vector measure by an standard procedure. Assuming that the solution of each individual equation of the system belongs to a Banach function space based on scalar measures we deduce, with natural conditions, that a solution of such system belongs to a Banach function space based on a vector measure. We also give an example of a system of non-linear Bernoulli equations and show the relation with an equation involving the integral operator.Mathematics Subject Classification. Primary 46G10; Secondary 47N20, 46N20.

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“…The main references on vector measures used in this paper are [9,24]. Let m : Σ → E be a vector measure, where E is a Banach space.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Mjommentioning

confidence: 99%

Section: Mjommentioning

confidence: 99%

Section: The Integral Operator Equationmentioning

confidence: 99%

See 3 more Smart Citations

On the Norm with Respect to Vector Measures of the Solution of an Infinite System of Ordinary Differential Equations

Galdames-Bravo

2014

Mediterr. J. Math.

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The weak topology on q‐convex Banach function spaces

Albesa

Calabuig

Pérez

2011

Mathematische Nachrichten

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Let X(μ) be a Banach function space. In this paper we introduce two new geometric notions, q‐convexity and weak q‐convexity associated to a subset S of the unit ball of the dual of X(μ), 1 ≤ q < ∞. We prove that in the general case both notions are not equivalent and we study the relation between them, showing that they can be used for describing the weak topology in these spaces. We define the canonical q‐concave weak topology τq on X(μ)—a topology generated by q‐concave seminorms—for obtaining our main result: A σ‐order continuous Banach function space X(μ) is q‐convex if and only if the following topological inclusions τw⊆τq⊆τ‖ · ‖ hold. As an application, in the last section we prove a suitable Maurey‐Rosenthal type factorization theorem for operators from a Banach function space X(μ) into a Banach space that holds under weaker assumptions on the q‐convexity requirements for X(μ).

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Summability in of a vector measure

Calabuig

Rodríguez

Pérez

2016

Mathematische Nachrichten

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We show a picture of the relations among different types of summability of series in the space L1false(mfalse) of integrable functions with respect to a vector measure m of relatively norm compact range. In order to do that, we study the class of the so‐called m‐1‐summing operators. We give several applications regarding the existence of copies of c0 in L1false(mfalse), as well as on m‐1‐summing operators which are weakly compact, Asplund or weakly precompact.

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Optimal Domain and Integral Extension of Operators

Cited by 165 publications

References 0 publications

On the Norm with Respect to Vector Measures of the Solution of an Infinite System of Ordinary Differential Equations

On the Norm with Respect to Vector Measures of the Solution of an Infinite System of Ordinary Differential Equations

The weak topology on q‐convex Banach function spaces

Summability in of a vector measure

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