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Properties (S) and (gS) for bounded linear operators. (English) Zbl 1466.47005

Summary: An operator \(T\) acting on a Banach space \(X\) obeys property (R) if \(\pi_a^0(T) = E^0(T)\), where \(\pi_a^0(T)\) is the set of all left poles of \(T\) of finite rank and \(E^0(T)\) is the set of all isolated eigenvalues of \(T\) of finite multiplicity. In this paper we introduce and study two new properties (S) and (gS) in connection with Weyl type theorems. Among other things, we prove that if \(T\) is a bounded linear operator acting on a Banach space, then \(T\) satisfies property (R) if and only if \(T\) satisfies property (S) and \(\pi^0(T) = \pi_a^0(T)\), where \(\pi^0(T)\) is the set of poles of finite rank. Also we show if \(T\) satisfies Weyl theorem, then \(T\) satisfies property (S). Analogous results for property (gS) are given. Moreover, these properties are also studied in the frame of polaroid operators.

MSC:

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