### 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 (μ_{1},...,μ_{n}) 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 language | English (US) |
---|---|

Pages (from-to) | 65-80 |

Number of pages | 16 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2007 |

### All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Geometry and Topology

## Fingerprint Dive into the research topics of 'Feynman's operational calculi: Spectral theory for noncommuting self-adjoint operators'. Together they form a unique fingerprint.

## Cite this

*Mathematical Physics Analysis and Geometry*,

*10*(1), 65-80. https://doi.org/10.1007/s11040-007-9021-8