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 $T\in \mathcal {M}$.References
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297, DOI 10.1017/CBO9780511526138
- Fernando Mayoral, Compact sets of compact operators in absence of $l^1$, Proc. Amer. Math. Soc. 129 (2001), no. 1, 79–82. MR 1784015, DOI 10.1090/S0002-9939-00-06007-X
- Theodore W. Palmer, Totally bounded sets of precompact linear operators, Proc. Amer. Math. Soc. 20 (1969), 101–106. MR 235425, DOI 10.1090/S0002-9939-1969-0235425-3
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