Riesz points of the spectrum of an element in a semisimple Banach algebra

Author:
Lynn D. Pearlman

Journal:
Trans. Amer. Math. Soc. **193** (1974), 303-328

MSC:
Primary 46H05; Secondary 47B05

DOI:
https://doi.org/10.1090/S0002-9947-1974-0346533-7

MathSciNet review:
0346533

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Abstract: Let *A* be a semisimple Banach algebra with unit element and let denote the socle of *A*. For an element *y* in *A*, let denote the operator of left [right] multiplication by *y* on *A*. The operational calculus and A. E. Taylor's theory of the ascent and descent of an operator *T* on *A* are used to show that the following conditions on a number in the spectrum of an element *x* in *A* are all equivalent. (1) is a pole of the resolvent mapping and the spectral idempotent *f*, for *x* at is in . (2) is invertible in *A* for some *c* in the closure of such that . (3) is invertible modulo the closure of and . (4) is invertible modulo the closure of and . Such numbers are called *Riesz points*. An element *x* is called a *Riesz element of A* if it is topologically nilpotent modulo the closure of . It is shown that *x* is a Riesz element if and only if every nonzero number in the spectrum of *x* is a Riesz point.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0346533-7

Keywords:
Semisimple Banach algebra,
Fredholm element socle,
ascent,
descent,
Riesz operator

Article copyright:
© Copyright 1974
American Mathematical Society