Equivalent conditions to the spectral decomposition property for closed operators

Abstract

Abstract.The spectral decomposition property has been instrumental in developing a local spectral theory for closed operators acting on a complex Banach space. This paper gives some necessary and sufficient conditions for a closed operator to possess the spectral decomposition property.In the monograph [3] and in a sequel of papers by the authors, a local spectral theory has been built for closed operators on the sole assumption of the spectral decomposition property. As an abstraction of Dunford's concept of "spectral reduction ' [2, p. 1927] and that of Bishop's "duality theory of type 3" [1], an operator T endowed with the spectral decomposition property produces a spectral decomposition of the underlying space, pertinent to any finite open cover of the spectrum o(T). In this paper we obtain some conditions equivalent to the spectral decomposition property. Some of them generalize results from[4]. PreliminariesGiven a Banach space X over the complex field C, we denote by C(X) the class of closed operators with domain DT and range in X, and we write Cd(X) for the subclass of the densely defined operators in C(X). For a subset Y of X, Y denotes the annihilator of Y in X* and for Zd',we use the symbol XZ for the preannihilator of Z in X. For the rest, the terminology and notation conform to that employed in [3].We shall adopt and adjust some concepts and ideas from Bishop's "duality theory of type 4" [1, §4]. A couple Ux and U2 of a bounded and an unbounded Cauchy domain, related by U2 = (Ux )c, are referred to as complementary simple sets. Wx and W2 are the sets of analytic functions from Ux to X and from U2 and X*, respectively, which vanish at oo. The seminorms \\f\\K¡ = max{||/(A)|| : /e Wx ,X € Kx,Kx (c Ux) is compact} and Hsll^ = max{||jy(A)|| : g € W2 ,X e K2 ,K2 (c U2) is compact}