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 | References | Similar Articles | Additional Information

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

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.