JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Some properties and applications of equicompact sets of operators

### Tom 181 / 2007

Studia Mathematica 181 (2007), 171-180 MSC: 47B07, 46G10. DOI: 10.4064/sm181-2-4

#### Streszczenie

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.

#### Autorzy

• E. SerranoDepartamento de Matemáticas
Campus Universitario del Carmen
Avda. de las Fuerzas Armadas s//n
E-21071 Huelva, Spain
e-mail
• C. PiñeiroDepartamento de Matemáticas
Campus Universitario del Carmen
Avda. de las Fuerzas Armadas s//n
E-21071 Huelva, Spain
e-mail
• J. M. DelgadoDepartamento de Matemáticas