On Decomposably Regular Operators

Christoph Schmoeger

Abstract: Let $X$ be a complex Banach space and $\calc{L}(X)$ the algebra of all bounded linear operators on $X$. $T\in\calc{L}(X)$ is said to be \Sle{decomposably regular} provided there is an operator $S$ such that $S$ is invertible in $\calc{L}(X)$ and $TST=T$. For each $T\in\calc{L}(X)$ 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 $F: U\to\calc{L}(X)$ 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 $\C\backslash\rho_{gr}(T)$.

Keywords: Decomposably regular operators; Fredholm operators.

Classification (MSC2000): 47A10, 47A53

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