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

Authors: G. W. Treese and E. P. Kelly
Journal: Proc. Amer. Math. Soc. 67 (1977), 123-128
MSC: Primary 47B30; Secondary 47A99
MathSciNet review: 0454712
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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.

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Keywords: Banach algebra, maximal group, boundary of maximal group, bounded operator, generalized Fredholm operator, projection, generalized inverse
Article copyright: © Copyright 1977 American Mathematical Society

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