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 ] 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 ∈ G [ V ] T \in G[V] to be in the boundary, B \mathcal {B} , of the maximal group, M \mathcal {M} , of invertible operators are determined. As a result, B ∩ G [ V ] \mathcal {B} \cap G[V] is the set of products of operators in M \mathcal {M} and operators in P \mathcal {P} , where P \mathcal {P} is the set of projections other than the identity operator and null operator.