FEYNMAN'S OPERATIONAL CALCULI FOR NONCOMMUTING OPERATORS: SPECTRAL THEORY

Abstract

In recent papers the authors presented the key ideas involved in their approach to Feynman's operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. The spectral properties of the disentangling maps are studied in this paper.