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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


Authors: E. P. Kelly and D. A. Hogan
Journal: Proc. Amer. Math. Soc. 32 (1972), 195-200
MSC: Primary 47.10
DOI: https://doi.org/10.1090/S0002-9939-1972-0290136-3
MathSciNet review: 0290136
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Abstract: Let V denote a Banach space over the reals, $ B[V]$ the bounded linear operators on V,f a linear functional defined on a complete subspace, (f), of V. A conservative operator is an element of the set $ {\mathcal{T}_f} = \{ T\vert T \in B[V], T((f)) \subseteq (f)\} $. In this setting this paper extends some of the results of a recent paper by Rhoades [Triangular summability methods and the boundary of the maximal group, Math. Z. 105 (1968), 284-290]. In this setting necessary and sufficient conditions are proven for $ T \in {\mathcal{T}_f}$ to be in the maximal group of invertible elements, $ \mathcal{M}$. Sufficient conditions are proven for $ T \in {\mathcal{T}_f}$ to be in the boundary, $ \mathcal{B}$, of $ \mathcal{M}$. It is proven that $ \mathcal{B}$ is a multiplicative semigroup and if (f) is nontrivial, then $ \mathcal{B}$ is nonconvex. Two questions raised in the paper by Rhoades were answered.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0290136-3
Article copyright: © Copyright 1972 American Mathematical Society