### 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 |
---|---|

Pages (from-to) | 65-80 |

Number of pages | 16 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2007 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Mathematical Physics Analysis and Geometry*,

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

**Feynman's operational calculi : Spectral theory for noncommuting self-adjoint operators.** / Jefferies, Brian; Johnson, Gerald W.; Nielsen, Lance.

Research output: Contribution to journal › Article

*Mathematical Physics Analysis and Geometry*, vol. 10, no. 1, pp. 65-80. https://doi.org/10.1007/s11040-007-9021-8

}

TY - JOUR

T1 - Feynman's operational calculi

T2 - Spectral theory for noncommuting self-adjoint operators

AU - Jefferies, Brian

AU - Johnson, Gerald W.

AU - Nielsen, Lance

PY - 2007/2

Y1 - 2007/2

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=34547235123&partnerID=8YFLogxK

U2 - 10.1007/s11040-007-9021-8

DO - 10.1007/s11040-007-9021-8

M3 - Article

VL - 10

SP - 65

EP - 80

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

SN - 1385-0172

IS - 1

ER -