Characterizations of closed decomposable operators
Shengwang Wang, Ivan N. Erdelyi · Illinois Journal of Mathematics · 1984
In extending the spectral theory beyond the class of spectral operators, one can no longer produce a spectral measure to represent the operator or to reveal its spectral structure.In this paper, we extend the use of some substi- tutes for the unavailable spectral measure, such as spectral capacity [1], [2-1, I-6] and spectral resolvent [3-1 for the spectral-theoretic study of closed oper- ators in an abstract Banach space.The program of the paper is as follows.After a preliminary section, we supplement the given closed operator with a weaker constituent than that of the spectral resolvent and obtain a new criterion for its spectral decomposi- tion (Corollary 2.3).In Section 3, we introduce a concept weaker than that of spectral capacity for a closed operator and again, we obtain a simpler description of its spectral decomposition (Theorem 3.3).Moreover, we extend a property of a bounded decomposable operator to the unbounded case (Corollary 3.5) and find conditions for a specific linear manifold occuring in the theory of spectral capacities, to be dense in the underlying space (Theorem 3.6).If not mentioned otherwise, throughout this paper T is an unbounded closed operator with domain Dr and range in a Banach space X over the complex field C. For a set S, g is the closure, S is the complement, cS is the boundary, d(2, S) is the distance from a point 2 6 C to S c C, and we write cov S for the collection of all finite open covers of S. If S is a subset of C, then the above mentioned topological constructs are referred to the topology G of C Without loss of generality, we assume that for $ c C, each }-o 6 cov S has, at most, one unbounded set Go.A set G c C is said to be a neighborhood of , in symbols G 6 V(R), if for r > 0 sufficiently large, We write S +/-for the annihilator of S c X in the dual space X*.We write t5and for the families of all open and closed subsets of C, respectively.Further, we denote by ffi K and K the collection of all relatively compact open and that of all compact subsets of C, respectively.N is the set of all positive integers.We use the notations a(T), p(T) and R(.; T) for the spectrum resolvent set and resolvent operator, respectively of T. If T has the single valued extension