Generalized Fredholm operators and the boundary of the maximal group of invertible operators

Abstract:

Let V denote an infinite dimensional Banach space over the complex field and let $G[V]$ denote the subset of bounded operators on V with the property that the null space has a closed complement and the range is closed, where the null space and range are proper subspaces of V. Necessary and sufficient conditions for $T \in G[V]$ to be in the boundary, $\mathcal {B}$, of the maximal group, $\mathcal {M}$, of invertible operators are determined. As a result, $\mathcal {B} \cap G[V]$ is the set of products of operators in $\mathcal {M}$ and operators in $\mathcal {P}$, where $\mathcal {P}$ is the set of projections other than the identity operator and null operator.