Feynman's operational calculi for noncommuting operators: The monogenic calculus
type: Item
instance of: scholarly article
title: Feynman's operational calculi for noncommuting operators: The monogenic calculus
zbMATH Open document ID: 1096.47501
DOI: 10.1007/BF03042315
published in: Advances in Applied Clifford Algebras
publication date: 2004-05-27T00:00:00Z
review text: This is one of several papers in which the authors present their approach to Feynman's operational calculi for \(n\)-tuples \(A=(A_1,\dots, A_n)\) of not necessarily commuting bounded linear operators \(A_j\) acting on a Banach space. Given an \(n\)-tuple \(\mu=(\mu_1,\dots, \mu_n)\) of continuous probability measures, an operator \(f_\mu(A)\) is determined whenever \(f\) is real analytic in a suitable subset of \(\mathbb{R}^n\). In essence, the measures determine the orders in which the operators act. If the operators satisfy certain exponential bounds, then \(f\) is only required to be smooth on a neighbourhood of a compact subset \(\gamma_\mu (A)\) of \(\mathbb{R}^n\) called the \(\mu\)-joint spectrum. If the operators satisfy the exponential bounds and \(f\) is left-monogenic on a neighbourhood of \(\gamma_\mu (A)\) in \(\mathbb{R}^{n+1}\), then \(f_\mu (A)\) is shown to be represented as a surface integral of functions with values in a Banach module over a Clifford algebra. This is a higher-dimensional analogue of the Riesz-Dunford calculus. Moreover, \(\gamma_\mu(A)\) is identified with the set of singularities of the kernel of the integral representation.
reviewed by: Alan J. Pryde
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 46H30
Mathematics Subject Classification ID: 47A25
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 2068488
zbMATH Keywords: Banach module
zbMATH Keywords: Clifford algebra
zbMATH Keywords: \(n\)-tuples of linear operators
zbMATH Keywords: \(\mu\)-joint spectrum
author: Brian Jefferies
author: G. W. Johnson
MaRDI profile type: Publication
cites work: The Weyl functional calculus
cites work: Q3310867
cites work: Q5584224
cites work: Q3244654
cites work: Feynman's operational calculi for noncommuting operators: definitions and elementary properties
cites work: Feynman's operational calculus for families of noncommuting operators: tensor products, ordered supports, and the disentangling of an exponential factor
cites work: FEYNMAN'S OPERATIONAL CALCULI FOR NONCOMMUTING OPERATORS: SPECTRAL THEORY
cites work: The Weyl calculus and Clifford analysis
cites work: Q4946766
cites work: Lacunas in the Support of the Weyl Calculus for Two Hermitian Matrices
cites work: Q4954180
cites work: Q4340880
cites work: Q3469767
cites work: Functions of Several Self-Adjoint Operators
full work available at URL: https://doi.org/10.1007/bf03042315
OpenAlex ID: W1965239337
description: scientific article; zbMATH DE number 2068488