$$\lambda $$ -Limited Sets in Banach and Dual Banach Spaces

Abstract

In this paper, we introduce the notions of \(\lambda \)-limited sets and \(\lambda \)-L-sets in a Banach space X and its dual \(X^*\) respectively, using the vector valued sequence spaces \(\lambda ^{w^*}(X^*)\) and \(\lambda ^{w}(X)\). We find characterizations for these sets in terms of absolutely \(\lambda \)-summing operators and investigate the relationship between \(\lambda \)-compact sets and \(\lambda \)-limited sets, with a particular focus on the crucial role played by a norm iteration property. We also consider \(\lambda \)-limited operators and show that this class is an operator ideal containing the ideal of \(\lambda \)-compact operators for a suitably restricted \(\lambda \). Furthermore, we define a generalized Gelfand-Philips property for Banach spaces corresponding to an abstract sequence space.