λ-Limited Sets in Banach and Dual Banach Spaces

In this paper, we introduce the notions of lambda-limited sets and lambda-L-sets in a Banach space X and its dual X-& lowast; respectively, using the vector valued sequence spaces lambda(w & lowast;)(X-& lowast;) 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.