Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Bounded, conservative, linear operators and the maximal group
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by E. P. Kelly and D. A. Hogan
Proc. Amer. Math. Soc. 32 (1972), 195-200
DOI: https://doi.org/10.1090/S0002-9939-1972-0290136-3

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 ${\mathcal {T}_f} = \{ T|T \in B[V], T((f)) \subseteq (f)\}$. 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.
References
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Bibliographic Information
  • © Copyright 1972 American Mathematical Society
  • 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