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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 $ {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.


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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

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