On the Localization of the Spectrum for Systems of Operators

Abstract

Let a = (a,.., a) be a commuting system of linear continuous operators on a complex Banach space X. We show that, for any x E X, the local analytic spectrum a(a, x) [1] is contained in the spectral hull of the local spectrum sp(a, x) [4]. Introduction. In this paper we shall define more accurately the relation between two notions of local spectrum for systems of operators. Let X be a complex Banach space and let a = (a, . . . , an) be a commuting system (n-tuple) of linear continuous operators on X. The simplest way to define the spectrum of an arbitrary element x E X, with respect to a, is the following. Consider the union p(a, x) of all open sets D C C' such that there exist n analytic functions fl, ... ,fn: Df, satisfying X7.= I(zi ai)f (z) x, z E D. Then the spectrum of x with respect to a is the set a(a, x) = C' p(a, x). This notion of local spectrum was used in [1] for the study of spectral decompositions dependent on functional calculi. In [4] the author has studied spectral decompositions not necessarily dependent on functional calculi and has found very useful to define another notion of local spectrum. Generally speaking, the spectrum of x with respect to a in our sense, denoted by sp(a, x), is the support of a certain differential form (or rather of a cohomology class of such forms). This notion was suggested by a new definition of the functional analytic calculus for several commuting operators, proposed in [7]. We have proved in [4] that sp(a, x) c a(a, x) for any x E X. The main result of the present paper is that a(a, x) is contained in the spectral hull of sp(a, x), for any x E X. 1. Preliminaries. We shall recall the definition of the Cauchy-Weil integral which will be our main tool in what follows (for details, see [7, ?3] and [4, Preliminaries]). Let X be a complex Banach space and a = (a,, .. an) be a commuting system of linear continuous operators on X. Denote by sp(a, X) the Taylor spectrum of a on X [6]. Received by the editors May 17, 1977. AMS (MOS) subject classifications (1970). Primary 47D99; Secondary 32C30, 32C35.