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Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let i : L ( X ) L ¹ ( X ) stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then T i : L ( X ) Y is a weakly compact operator. Moreover, we obtain that if T: L¹(X)...

Dieudonné-type theorems for lattice group-valued k -triangular set functions

Antonio Boccuto, Xenofon Dimitriou (2019)

Kybernetika

Some versions of Dieudonné-type convergence and uniform boundedness theorems are proved, for k -triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case. Furthermore, we pose some open problems.

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