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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Equicompact sets of operators defined on Banach spaces
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by E. Serrano, C. Piñeiro and J. M. Delgado PDF
Proc. Amer. Math. Soc. 134 (2006), 689-695 Request permission

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 $\|Tx\|\leq \sup _n|x_n^*(x)|$ 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 $\|Tx_i\|\leq (1+\varepsilon )\|Sx_i\|$ 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
  • Received by editor(s): April 20, 2004
  • Published electronically: October 17, 2005
  • Communicated by: Jonathan M. Borwein
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 689-695
  • MSC (2000): Primary 47B07
  • DOI: https://doi.org/10.1090/S0002-9939-05-08338-3
  • MathSciNet review: 2180885