Some properties and applications of equicompact sets of operators

E. Serrano, C. Piñeiro, and J. M. Delgado. "Some properties and applications of equicompact sets of operators." Studia Mathematica 181.2 (2007): 171-180. <http://eudml.org/doc/285112>.

@article{E2007,
abstract = {Let X and Y be Banach spaces. A subset M of (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 (xₙ) in X has a subsequence $(x_\{k(n)\})ₙ$ such that $(Tx_\{k(n)\})ₙ$ is uniformly convergent for T ∈ 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 $ℳ_\{c\}(ℱ,X)$, the Banach space of all (finitely additive) vector measures (with compact range) from a field ℱ of sets into X endowed with the semivariation norm.},
author = {E. Serrano, C. Piñeiro, J. M. Delgado},
journal = {Studia Mathematica},
keywords = {compact operators; equicompact sets of operators; collectively compact set; vector measures; Ascoli's theorem},
language = {eng},
number = {2},
pages = {171-180},
title = {Some properties and applications of equicompact sets of operators},
url = {http://eudml.org/doc/285112},
volume = {181},
year = {2007},
}

TY - JOUR
AU - E. Serrano
AU - C. Piñeiro
AU - J. M. Delgado
TI - Some properties and applications of equicompact sets of operators
JO - Studia Mathematica
PY - 2007
VL - 181
IS - 2
SP - 171
EP - 180
AB - Let X and Y be Banach spaces. A subset M of (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 (xₙ) in X has a subsequence $(x_{k(n)})ₙ$ such that $(Tx_{k(n)})ₙ$ is uniformly convergent for T ∈ 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 $ℳ_{c}(ℱ,X)$, the Banach space of all (finitely additive) vector measures (with compact range) from a field ℱ of sets into X endowed with the semivariation norm.
LA - eng
KW - compact operators; equicompact sets of operators; collectively compact set; vector measures; Ascoli's theorem
UR - http://eudml.org/doc/285112
ER -