Bounded, conservative, linear operators and the maximal group
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- by E. P. Kelly and D. A. Hogan PDF
- Proc. Amer. Math. Soc. 32 (1972), 195-200 Request permission
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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- B. E. Rhoades, Triangular summability methods and the boundary of the maximal group, Math. Z. 105 (1968), 284–290. MR 228882, DOI 10.1007/BF01125969
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Additional 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