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Spectral continuity using \(\nu\)-convergence. (English) Zbl 1325.47032

Summary: Let \(T, T_n\), \(n \in \mathbb{N}\), be bounded linear operators defined on a Banach space \(X\) such that \(\{T_n \}\) converges to \(T\) in \(\nu\)-convergence (this new mode of convergence was observed by Mario Ahues). In this paper, sufficient conditions are given for the convergence \(\gamma(T_n) \to \gamma(T)\), where \(\gamma \in \{\sigma, \sigma_{\mathit{ap}} \}\). Also we give some conditions for a bounded operator \(S\) in order to have the stability of convergence: \(\gamma(T_n + S) \to \gamma(T + S)\). Among other things, we show that, if 0 is an accumulation point of \(\sigma(T)\), then \(\sigma(T_n) \to \sigma(T)\) when \(T_n, T\) satisfy the growth condition \((G_1)\).

MSC:

47A58 Linear operator approximation theory
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