   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 $\|Tx\|\leq \sup _n|x_n^*(x)|$ 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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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