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 on a Banach space , there is an extension of such that extends in a natural way to an operator on , and the spectrum of is the approximate point spectrum of . 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 containing elements and such that in no extension of are the residual spectra of and eliminated simultaneously.

**[1]**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**, 10.1090/S0002-9947-1984-0760982-0**[2]**B. Bollobás,*Adjoining inverses to commutative Banach algebras*, Algebras in Analysis (J. H. Williamson, ed.), Academic Press, New York, 1975, pp. 256-257.**[3]**G. E. Shilov,*On normed rings with one generator*, Mat. Sb.**21**(**63**) (1947), 25-46.**[4]**Richard Arens,*Linear topological division algebras*, Bull. Amer. Math. Soc.**53**(1947), 623–630. MR**0020987**, 10.1090/S0002-9904-1947-08857-1**[5]**Béla Bollobás,*Adjoining inverses to commutative Banach algebras*, Trans. Amer. Math. Soc.**181**(1973), 165–174. MR**0324418**, 10.1090/S0002-9947-1973-0324418-9**[6]**Béla Bollobás,*Best possible bounds of the norms of inverses adjoined to normed algebras*, Studia Math.**51**(1974), 87–96. MR**0348502****[7]**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**, 10.1007/BF02766221

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Additional Information

DOI:
http://dx.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