Equicompact sets of operators defined on Banach spaces
Abstract. Let X and Y be Banach spaces. We say that a set M⊂ K(X, Y ) (K(X, Y ) denotes the space of all compact operators from X into Y ) is equicompact if there exists a null sequence (x∗ n)n in X∗ such that Tx ≤ supn |x∗ n(x)| for all x ∈ X and all T ∈ M. It is easy to show that collectively comp...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2006 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/163390 |
| Acceso en línea: | https://hdl.handle.net/11441/163390 https://doi.org/10.1090/S0002-9939-05-08338-3 |
| Access Level: | acceso abierto |
| Palabra clave: | Compact operators Equicompact set Collectively compact set |
| Sumario: | Abstract. Let X and Y be Banach spaces. We say that a set M⊂ K(X, Y ) (K(X, Y ) denotes the space of all compact operators from X into Y ) is equicompact if there exists a null sequence (x∗ n)n in X∗ such that Tx ≤ supn |x∗ n(x)| for all x ∈ X and all T ∈ M. It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: Mis equicompact iff M∗ = {T∗ : T ∈ M} is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set M ⊂ K(X, Y ) is equicompact iff each bounded sequence (xn)n in X has a subsequence (xk(n))n such that (Txk(n))n is a converging sequence uniformly for T ∈ M; 2) if Y does not have finite cotype and M⊂ K(X, Y ) is a maximal equicompact set, then, given ε > 0 and a finite set {x1, . . . ,xn} in X, there is an operator S ∈ M such that Txi ≤ (1 + ε) Sxi for i = 1, . . . , n and all T ∈ M. |
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