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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Bounded, conservative, linear operators and the maximal group. II

Authors: E. P. Kelly and D. A. Hogan
Journal: Proc. Amer. Math. Soc. 38 (1973), 298-302
MSC: Primary 46L20
MathSciNet review: 0313832
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Abstract: Let $ V$ denote an infinite dimensional Banach space over the complex field, $ B[V]$ the bounded linear operators on $ V$ and $ F$ a closed subspace of $ V$. An element of $ {\mathcal{T}_F} = \{ T\vert T \in B[V],T(F) \subseteq F\} $ is called a conservative operator. Some sufficient conditions for $ T \in {\mathcal{T}_F}$ to be in the boundary, $ \mathcal{B}$, of the maximal group, $ \mathcal{M}$, of invertible elements are determined. For example, if $ T \in {\mathcal{T}_F}$, is such that (i) $ V$ is the topological direct sum of $ \mathcal{R}(T)$ and $ N(T) \ne \{ \theta \} $, (ii) $ T$ is an automorphism on $ \mathcal{R}(T) \cap F$, then $ T \in \mathcal{B}$. Also, the complement of the closure of $ \mathcal{M}$ is discussed. This is an extension of another paper by the same authors [6].

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Keywords: Banach algebra, maximal group, boundary of maximal group, bounded operator, conservative operator, projection, quasi-compact operator
Article copyright: © Copyright 1973 American Mathematical Society

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