Feynman's operational calculi: Spectral theory for noncommuting self-adjoint operators

Brian Jefferies, Gerald W. Johnson, Lance Nielsen

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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 A 1,..., A n and associated continuous Borel probability measures μ 1, ⋯, μ n on [0,1]. Fix A 1,..., A n . Then each choice of an n-tuple (μ1,...,μn) of measures determines one of Feynman's operational calculi acting on a certain Banach algebra of analytic functions even when A 1, ..., A n 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.

Original languageEnglish
Pages (from-to)65-80
Number of pages16
JournalMathematical Physics Analysis and Geometry
Volume10
Issue number1
DOIs
Publication statusPublished - Feb 2007

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All Science Journal Classification (ASJC) codes

  • Physics and Astronomy (miscellaneous)
  • Mathematical Physics
  • Mathematics(all)

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