Meromorphic functions of elements of a commutative Banach algebra

Abstract

Introduction. We are concerned here with applications of the author's paper The Riemann sphere of a commutative Banach algebra [7] to operational calculus and to abstract analytic function theory. Let A be a commutative complex Banach algebra with identity, A,, its Riemann sphere, C the complex plane, and Ccx, the classical Riemann sphere, i.e. the extended complex plane. In ?0, for the reader's convenience, we present the relevant theorems and definitions of [7], together with some minor additions. No proofs are given there; they are either in [7], or straightforward. In ?1 h(p) is defined, where p E A,, and h is CO0-valued and meromorphic on a neighborhood of the spectrum of p in Ccx,. (Thus in particular, h(p) is defined when h is a rational function.) The sum h,(p) + h2(p) is seen to be defined in A 00 iff h, + h2 have no common on the spectrum of p; then h1(p)+h2(p)=(h,+h2)(p). Similarly, the product h,(p)h2(p) is seen to be defined in A,, iff there is no point in the spectrum of p which is simultaneously a of one of the hi and a zero of the other; then h1(p)h2(p)=(h,h2)(p). The spectral mapping theorem spectrum h(p) = h(spectrum p) is proved. A straightforward consequence of this operational calculus is that each meromorphic h: C -* CO,O lifts uniquely to a meromorphic h: A A,,. The chief result of ?2 is a generalization of the Mittag-Leffler theorem. We show that given a sequence zn in C with no limit point, and a suitable principal part of a Laurent expansion (with coefficients in A) at each zn, the classical MittagLeffler construction yields a meromorphic p: A A,, so that q4 C is A-valued and analytic except at the zn, P IC has poles at the zn, the principal part of the Laurent expansion of p I C at each zn is the one given above, and p(a) lies in A iff the spectrum of a contains none of the zn. Useful in the proof is the quotient function lemma [7, p. 22]. As a corollary, the Mittag-Leffler decomposition of a meromorphic h: C -* Co. is seen to remain valid for the meromorphic extension h: A -* A,,. To accomplish this generalization, it is necessary to redefine the notion of pole of analytic A-valued function of a complex variable. The standard definition [4, p. 236], is an isolated singularity off at which the principal part of the Laurent expansion has finitely many nonzero terms. Our definition is isolated