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On decomposably regular operators. (English) Zbl 0891.47004
Summary: Let \(X\) be a complex Banach space and \({\mathcal L}(X)\) the algebra of all bounded linear operators on \(X\). \(T\in{\mathcal L}(X)\) is said to be decomposably regular provided there is an operator \(S\) such that \(S\) is invertible in \({\mathcal L}(X)\) and \(TST= T\). For each \(T\in{\mathcal L}(X)\), we introduce the following subset \(\rho_{gr}(T)\) of the resolvent set of \(T:\mu\in\rho_{gr}(T)\) if and only if there is a neighbourhood \(U\) of \(\mu\) and a holomorphic function \(F:U\to{\mathcal L}(X)\) such that \(F(\lambda)\) is invertible for all \(\lambda\in U\) and \((T-\lambda)F(\lambda)(T- \lambda)= T-\lambda\) on \(U\). In this note, we determine the interior points of the class of decomposably regular operators and we prove a spectral mapping theorem for \(\mathbb{C}\backslash\rho_{gr}(T)\).

47A10 Spectrum, resolvent
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A53 (Semi-) Fredholm operators; index theories
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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