Sobre operadores y sucesiones en espacios de Banach separables

Abstract

Let B and X be Banach spaces, let B be reflexive, and let (an) be a sequence in B. The author shows that inf{∑|fan|:f∈B*,||f||=1}>0 if and only if the subspace {f∈B*:∑|fan|<∞} contains no closed infinite-dimensional subspace of B*, and in that case we have, for any bounded operator T:B→X,(i)T* is strictly singular whenever ∑Tan is weakly unconditionally Cauchy, and (ii) T is compact whenever ∑||Tan||<∞.