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 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
Feynman's operational calculi : Spectral theory for noncommuting self-adjoint operators. / Jefferies, Brian; Johnson, Gerald W.; Nielsen, Lance.
In: Mathematical Physics Analysis and Geometry, Vol. 10, No. 1, 02.2007, p. 65-80.Research output: Contribution to journal › Article
}
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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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 -