Spectrum reducing extension for one operator on a Banach space
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- by C. J. Read PDF
- Trans. Amer. Math. Soc. 308 (1988), 413-429 Request permission
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.References
- C. J. Read, Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element, Trans. Amer. Math. Soc. 286 (1984), no. 2, 715–725. MR 760982, DOI 10.1090/S0002-9947-1984-0760982-0 B. Bollobás, Adjoining inverses to commutative Banach algebras, Algebras in Analysis (J. H. Williamson, ed.), Academic Press, New York, 1975, pp. 256-257. G. E. Shilov, On normed rings with one generator, Mat. Sb. 21(63) (1947), 25-46.
- Richard Arens, Linear topological division algebras, Bull. Amer. Math. Soc. 53 (1947), 623–630. MR 20987, DOI 10.1090/S0002-9904-1947-08857-1
- Béla Bollobás, Adjoining inverses to commutative Banach algebras, Trans. Amer. Math. Soc. 181 (1973), 165–174. MR 324418, DOI 10.1090/S0002-9947-1973-0324418-9
- Béla Bollobás, Best possible bounds of the norms of inverses adjoined to normed algebras, Studia Math. 51 (1974), 87–96. MR 348502, DOI 10.4064/sm-51-2-87-96
- C. J. Read, Extending an operator from a Hilbert space to a larger Hilbert space, so as to reduce its spectrum, Israel J. Math. 57 (1987), no. 3, 375–380. MR 889985, DOI 10.1007/BF02766221
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 308 (1988), 413-429
- MSC: Primary 47A20; Secondary 46H05, 47A10
- DOI: https://doi.org/10.1090/S0002-9947-1988-0946450-5
- MathSciNet review: 946450