# zbMATH — the first resource for mathematics

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 $$\mathbb{C}\backslash\rho_{gr}(T)$$.

##### MSC:
 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.
Full Text: