\(\lambda \)-Limited Sets in Banach and Dual Banach Spaces

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