Feynman’s Operational Calculi: Spectral Theory for Noncommuting Self-adjoint Operators

Abstract

The spectral theorem for commuting self-adjoint operators along with the associated functional (or operational) calculus is among the most useful and beautiful results of analysis. It is well known that forming a functional calculus for noncommuting self-adjoint operators is far more problematic. The central result of this paper establishes a rich functional calculus for any finite number of noncommuting (i.e. not necessarily commuting) bounded, self-adjoint operators A1,..., An and associated continuous Borel probability measures μ1, ⋯, μn on [0,1]. Fix A1,..., An. Then each choice of an n-tuple \(\left(\mu_1,\ldots,\mu_n\right)\) of measures determines one of Feynman’s operational calculi acting on a certain Banach algebra of analytic functions even when A1, ..., An are just bounded linear operators on a Banach space. The Hilbert space setting along with self-adjointness allows us to extend the operational calculi well beyond the analytic functions. Using results and ideas drawn largely from the proof of our main theorem, we also establish a family of Trotter product type formulas suitable for Feynman’s operational calculi.