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

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.