Some properties and applications of equicompact sets of operators

E. Serrano¹ ; C. Piñeiro¹ ; J. M. Delgado¹

¹Departamento de Matemáticas Facultad de Ciencias Experimentales Campus Universitario del Carmen Avda. de las Fuerzas Armadas s//n E-21071 Huelva, Spain

Studia Mathematica, Tome 181 (2007) no. 2, pp. 171-180

Résumé

Let $X$ and $Y$ be Banach spaces. A subset ${\rm M}$ of ${\cal K}(X,Y)$ (the vector space of all compact operators from $X$ into $Y$ endowed with the operator norm) is said to be equicompact if every bounded sequence $(x_n)$ in $X$ has a subsequence $(x_{k(n)})_n$ such that $(Tx_{k(n)})_n$ is uniformly convergent for $T\in{\rm M}$. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness criterion in ${\cal M}_{\rm c}({\cal F},X)$, the Banach space of all (finitely additive) vector measures (with compact range) from a field ${\cal F}$ of sets into $X$ endowed with the semivariation norm.

Détail

DOI : 10.4064/sm181-2-4

Keywords: banach spaces subset cal vector space compact operators endowed operator norm said equicompact every bounded sequence has subsequence uniformly convergent study relationship between concept notion uniformly completely continuous set applications among other results obtain generalization classical ascoli theorem compactness criterion cal cal banach space finitely additive vector measures compact range field cal sets endowed semivariation norm