λ-Limited Sets in Banach and Dual Banach Spaces.

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