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On new strong versions of Browder type theorems. (English) Zbl 06893167

Summary: An operator \(T\) acting on a Banach space \(X\) satisfies the property \((UW_\varPi)\) if \(\sigma_a(T) \smallsetminus \sigma_{SF_+^-}(T) = \varPi(T)\), where \(\sigma_a(T)\) is the approximate point spectrum of \(T\), \(\sigma_{SF_+^-}(T)\) is the upper semi-Weyl spectrum of \(T\) and \(\varPi(T)\) the set of all poles of \(T\). In this paper we introduce and study two new spectral properties, namely \((V_\varPi)\) and \((V_{\varPi_a})\), in connection with Browder type theorems introduced in [1], [2], [3] and [4]. Among other results, we have that \(T\) satisfies property \((V_\varPi)\) if and only if \(T\) satisfies property \((UW_\varPi)\) and \(\sigma(T) = \sigma_a(T)\).

MSC:

47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A53 (Semi-) Fredholm operators; index theories
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