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Some properties and applications of equicompact sets of operators

100%

Serrano E. , Piñeiro C. , Delgado J.

Studia Mathematica

2007

tom 181

nr 2

171-180

Let X and Y be Banach spaces. A subset M of 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 (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.

Operators whose adjoints are quasi p-nuclear

100%

Delgado J. , Piñeiro C. , Serrano E.

Studia Mathematica

2010

tom 197

nr 3

291-304

For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xₙ) in X with $K ⊆ {∑ₙαₙxₙ: (αₙ) ∈ B_{ℓ_{p'}}}$. We prove that an operator T: X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T* is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.