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Duality of measures of non-${\mathcal A}$-compactness

Volume 229 / 2015

Juan Manuel Delgado, Cándido Piñeiro Studia Mathematica 229 (2015), 95-112 MSC: Primary 47L20, 47B10; Secondary 47H08. DOI: 10.4064/sm7984-1-2016 Published online: 4 January 2016

Abstract

Let ${\mathcal A}$ be a Banach operator ideal. Based on the notion of ${\mathcal A}$-compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non-${\mathcal A}$-compactness of an operator. We consider a map $\chi _{\mathcal A}$ (respectively, $n_{\mathcal A}$) acting on the operators of the surjective (respectively, injective) hull of ${\mathcal A}$ such that $\chi _{{\mathcal A}}(T)=0$ (respectively, $n_{\mathcal A}(T)=0$) if and only if the operator $T$ is ${\mathcal A}$-compact (respectively, injectively ${\mathcal A}$-compact). Under certain conditions on the ideal ${\mathcal A}$, we prove an equivalence inequality involving $\chi _{\mathcal A}(T^*)$ and $n_{{\mathcal A}^d}(T)$. This inequality provides an extension of a previous result stating that an operator is quasi $p$-nuclear if and only if its adjoint is $p$-compact in the sense of Sinha and Karn.

Authors

  • Juan Manuel DelgadoDepartamento de Matemática Aplicada I
    Escuela Técnica Superior de Arquitectura Avenida Reina Mercedes, 2
    41012 Seville, Spain
    e-mail
  • Cándido PiñeiroDepartamento de Matemáticas
    Facultad de Ciencias Experimentales
    Campus Universitario de El Carmen
    21071 Huelva, Spain
    e-mail

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