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

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.

**[1]**Bruce A. Barnes,*Modular annihilator algebras*, Canad. J. Math.**18**(1966), 566–578. MR**0194471****[2]**Bruce Alan Barnes,*A generalized Fredholm theory for certain maps in the regular representations of an algebra*, Canad. J. Math.**20**(1968), 495–504. MR**0232208****[3]**Bruce Alan Barnes,*The Fredholm elements of a ring*, Canad. J. Math.**21**(1969), 84–95. MR**0237542****[4]**Felix E. Browder,*On the spectral theory of elliptic differential operators. I*, Math. Ann.**142**(1960/1961), 22–130. MR**0209909****[5]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[6]**David Kleinecke,*Almost-finite, compact, and inessential operators*, Proc. Amer. Math. Soc.**14**(1963), 863–868. MR**0155197**, 10.1090/S0002-9939-1963-0155197-5**[7]**David Lay,*Characterizations of the essential spectrum of F. E. Browder*, Bull. Amer. Math. Soc.**74**(1968), 246–248. MR**0221318**, 10.1090/S0002-9904-1968-11905-6**[8]**W. Pfaffenberger,*Operator algebras and related topics*, Doctoral Dissertation, University of Oregon, Eugene, Ore., 1969.**[9]**Charles E. Rickart,*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101****[10]**A. F. Ruston,*Operators with a Fredholm theory*, J. London Math. Soc.**29**(1954), 318–326. MR**0062345****[11]**Martin Schechter,*On the essential spectrum of an arbitrary operator. I*, J. Math. Anal. Appl.**13**(1966), 205–215. MR**0188798****[12]**Angus E. Taylor,*Introduction to functional analysis*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR**0098966****[13]**Angus E. Taylor,*Theorems on ascent, descent, nullity and defect of linear operators*, Math. Ann.**163**(1966), 18–49. MR**0190759****[14]**T. T. West,*Riesz operators in Banach spaces*, Proc. London Math. Soc. (3)**16**(1966), 131–140. MR**0193522**

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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