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Let A be a semisimple Banach algebra with unit element and let SA denote the socle of A. For an element y in A, let Ly [RyI denote the operator of left [right] multiplication by y on A. The operational calculus and A. E. Taylor's theory of the ascent a(T) and descent 3(T) of an operator T on A are used to show that the following conditions on a number A in the spectrum of an element x in A are all equivalent. (1) A is a pole of the resolvent mapping z (z -x)1 and the spectral idempotent f, for x at A is in SA. (2) A x c is invertible in A for some c in the closure of SA such that cx xc. (3) A -x is invertible modulo the closure of SA and 0 < ac(L(Ax)) -(L(Ax)) < oo. (4) A-x is invertible modulo the closure of SA and 0 < a(R(x))= S(R(x)) = a(L x)) = S(L(A-x)) < oo. Such numbers A are called Riesz points. An element x i s called a Riess element of A if it is topologically nilpotent modulo the closure of SA. 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. Introduction. Previous authors ([41, [141, [71) have studied poles of the resolvent with finite rank for operators on Banach space. Many of their results are generalized here to study the spectra of elements in a semisimple Banach algebra A. In ?1, the generalized Fredholm theory [31 is outlined. It is then used to give a necessary and sufficient condition for an arbitrary element x in A to have an associated element s in the socle such that x + s is invertible. In ?2, numbers which are poles of the resolvent mapping are considered. The theorems from the first two sections are used in ?3 to give several characterizations of Riesz points of the spectrum of an element. Finally in ?4 the Riesz elements of A, the analogous to well-known Riesz operators on Banach space, are considered. Presented to the Society, January 26, 1973 under the title Riesz points of the spectrum; received by the editors February 23, 1973. AMS (MOS) subject classifications (1970). Primary 46H05; Secondary 47B30, 47A55.

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