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 | References | Similar Articles | Additional Information

Abstract: Let *A* be a semisimple Banach algebra with unit element and let ${S_A}$ denote the socle of *A*. For an element *y* in *A*, let ${L_y}[{R_y}]$ denote the operator of left [right] multiplication by *y* on *A*. The operational calculus and A. E. Taylor’s theory of the ascent $\alpha (T)$ and descent $\delta (T)$ of an operator *T* on *A* are used to show that the following conditions on a number $\lambda$ in the spectrum of an element *x* in *A* are all equivalent. (1) $\lambda$ is a pole of the resolvent mapping $z \to {(z - x)^{ - 1}}$ and the spectral idempotent *f*, for *x* at $\lambda$ is in ${S_A}$. (2) $\lambda - x - c$ is invertible in *A* for some *c* in the closure of ${S_A}$ such that $cx = xc$. (3) $\lambda - x$ is invertible modulo the closure of ${S_A}$ and $0 < \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) < \infty$. (4) $\lambda - x$ is invertible modulo the closure of ${S_A}$ and $0 < \alpha ({R_{(\lambda - x)}}) = \delta ({R_{(\lambda - x)}}) = \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) < \infty$. Such numbers $\lambda$ are called *Riesz points*. An element *x* is called a *Riesz element of A* if it is topologically nilpotent modulo the closure of ${S_A}$. 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.

- Bruce A. Barnes,
*Modular annihilator algebras*, Canadian J. Math.**18**(1966), 566–578. MR**194471**, DOI https://doi.org/10.4153/CJM-1966-055-6 - Bruce Alan Barnes,
*A generalized Fredholm theory for certain maps in the regular representations of an algebra*, Canadian J. Math.**20**(1968), 495–504. MR**232208**, DOI https://doi.org/10.4153/CJM-1968-048-2 - Bruce Alan Barnes,
*The Fredholm elements of a ring*, Canadian J. Math.**21**(1969), 84–95. MR**237542**, DOI https://doi.org/10.4153/CJM-1969-009-1 - Felix E. Browder,
*On the spectral theory of elliptic differential operators. I*, Math. Ann.**142**(1960/61), 22–130. MR**209909**, DOI https://doi.org/10.1007/BF01343363 - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - David Kleinecke,
*Almost-finite, compact, and inessential operators*, Proc. Amer. Math. Soc.**14**(1963), 863–868. MR**155197**, DOI https://doi.org/10.1090/S0002-9939-1963-0155197-5 - David Lay,
*Characterizations of the essential spectrum of F. E. Browder*, Bull. Amer. Math. Soc.**74**(1968), 246–248. MR**221318**, DOI https://doi.org/10.1090/S0002-9904-1968-11905-6
W. Pfaffenberger, - 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** - A. F. Ruston,
*Operators with a Fredholm theory*, J. London Math. Soc.**29**(1954), 318–326. MR**62345**, DOI https://doi.org/10.1112/jlms/s1-29.3.318 - Martin Schechter,
*On the essential spectrum of an arbitrary operator. I*, J. Math. Anal. Appl.**13**(1966), 205–215. MR**188798**, DOI https://doi.org/10.1016/0022-247X%2866%2990085-0 - Angus E. Taylor,
*Introduction to functional analysis*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR**0098966** - Angus E. Taylor,
*Theorems on ascent, descent, nullity and defect of linear operators*, Math. Ann.**163**(1966), 18–49. MR**190759**, DOI https://doi.org/10.1007/BF02052483 - T. T. West,
*Riesz operators in Banach spaces*, Proc. London Math. Soc. (3)**16**(1966), 131–140. MR**193522**, DOI https://doi.org/10.1112/plms/s3-16.1.131

*Operator algebras and related topics*, Doctoral Dissertation, University of Oregon, Eugene, Ore., 1969.

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

Article copyright:
© Copyright 1974
American Mathematical Society