Decomposability and Structure of Semigroups of Nonnegative r-Potent Operators on L2 (X)

Abstract

An operator A on a separable Hilbert space L2 (X) (where X is a separable locally compact Hausdorff Space) is called an r-potent operator if Ar = A for some natural number r [1]. For the special case of r = 2, an r-potent operator reduces to an idempotent operator. The decomposability of general r-potent operators is an unsolved problem of operator theory. The decomposability of the special case of nonnegative r-potent operators was established by the authors in [2]. In this paper, we extend the results of [2] to semigroups of nonnegative r-potent operators and derive precise conditions under which a semigroup of nonnegative r-potent operators is decomposable. Further, we provide key results related to the structure of single decomposable nonnegative r-potent operators and use these results to derive the structure of decomposable semigroups of r-potents.