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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spectrum reducing extension for one operator on a Banach space


Author: C. J. Read
Journal: Trans. Amer. Math. Soc. 308 (1988), 413-429
MSC: Primary 47A20; Secondary 46H05, 47A10
MathSciNet review: 946450
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Abstract: In this paper we show that, given an operator $ T$ on a Banach space $ X$, there is an extension $ Y$ of $ X$ such that $ T$ extends in a natural way to an operator $ {T^ \sim }$ on $ Y$, and the spectrum of $ {T^ \sim }$ is the approximate point spectrum of $ T$. This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra $ L$ containing elements $ {g_1}$ and $ {g_2}$ such that in no extension $ L' $ of $ L$ are the residual spectra of $ {g_1}$ and $ {g_{_2}}$ eliminated simultaneously.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0946450-5
Keywords: Extension, inverse producing, spectrum reducing, functional calculus, transfinite induction
Article copyright: © Copyright 1988 American Mathematical Society