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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Equicompact sets of operators defined on Banach spaces


Authors: E. Serrano, C. Piñeiro and J. M. Delgado
Journal: Proc. Amer. Math. Soc. 134 (2006), 689-695
MSC (2000): Primary 47B07
DOI: https://doi.org/10.1090/S0002-9939-05-08338-3
Published electronically: October 17, 2005
MathSciNet review: 2180885
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Abstract: Let $ X$ and $ Y$ be Banach spaces. We say that a set $ \mathcal{M}\subset\mathcal{K}(X,Y)$ $ (\mathcal{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 $ \Vert Tx\Vert\leq\sup_n\vert x_n^*(x)\vert$ for all $ x\in X$ and all $ T\in\mathcal{M}$. It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: $ \mathcal{M}$ is equicompact iff $ \mathcal{M}^*=\{T^*\colon T\in\mathcal{M}\}$ is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set $ \mathcal{M}\subset\mathcal{K}(X,Y)$ is equicompact iff each bounded sequence $ (x_n)_n$ in $ X$ has a subsequence $ (x_{k(n)})_n$ such that $ (Tx_{k(n)})_n$ is a converging sequence uniformly for $ T\in\mathcal{M}$; 2) if $ Y$ does not have finite cotype and $ \mathcal{M}\subset\mathcal{K}(X,Y)$ is a maximal equicompact set, then, given $ \varepsilon>0$ and a finite set $ \{x_1,\ldots,x_n\}$ in $ X$, there is an operator $ S\in\mathcal{M}$ such that $ \Vert Tx_i\Vert\leq(1+\varepsilon)\Vert Sx_i\Vert$ for $ i=1, \ldots,n$ and all $ T\in\mathcal{M}$.


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Additional Information

E. Serrano
Affiliation: Departamento de Matemáticas, Facultad de Ciencias Experimentales, Campus Universitario del Carmen, Avda. de las Fuerzas Armadas s/n, 21071 Huelva, Spain
Email: eserrano@uhu.es

C. Piñeiro
Affiliation: Departamento de Matemáticas, Facultad de Ciencias Experimentales, Campus Universitario del Carmen, Avda. de las Fuerzas Armadas s/n, 21071 Huelva, Spain
Email: candido@uhu.es

J. M. Delgado
Affiliation: Departamento de Matemáticas, Facultad de Ciencias Experimentales, Campus Universitario del Carmen, Avda. de las Fuerzas Armadas s/n, 21071 Huelva, Spain
Email: jmdelga@uhu.es

DOI: https://doi.org/10.1090/S0002-9939-05-08338-3
Keywords: Compact operators, equicompact set, collectively compact set
Received by editor(s): April 20, 2004
Published electronically: October 17, 2005
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.