PORTUGALIAE MATHEMATICA Vol. 54, No. 1, pp. 4150 (1997) 

On Decomposably Regular OperatorsChristoph SchmoegerOn Decomposably Regular OperatorsAbstract: 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 Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1997 Sociedade Portuguesa de Matemática
