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Some properties and applications of equicompact sets of operators

E. Serrano; C. Piñeiro; J. M. Delgado — 2007

Studia Mathematica

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 $(T x_{k (n)}) ₙ$ 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...

Operators whose adjoints are quasi p-nuclear

J. M. Delgado; C. Piñeiro; E. Serrano — 2010

Studia Mathematica

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 $K \subseteq \sum ₙ α ₙ x ₙ : (α ₙ) \in B_{ℓ_{p^{'}}}$ . 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.

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