Spectral properties of noncommuting operators
type: Item
instance of: scholarly article
title: Spectral properties of noncommuting operators
zbMATH Open document ID: 1056.47002
published in: Lecture Notes in Mathematics
publication date: 2004-12-14T00:00:00Z
review text: A functional calculus for an \(n\)-tuple \((A_1,\ldots,A_n)\) of operators associates to certain scalar functions \(f\) of \(n\) variables an operator \(f(A_1,\ldots,A_n)\). If the operators do not commute, there is no canonical definition even for the simple function \(f(x_1,x_2)=x_1x_2\). For the Weyl calculus which is motivated from quantum mechanics and defined by a Fourier transform in Banach spaces, the symmetric average \((A_1A_2+A_2A_1)/2\) is chosen. However, even in rather simple situations the Weyl calculus and even its support (considering the Weyl calculus as an operator-valued tempered distribution) can be very hard to determine. This support is one of several possible natural definitions for the spectrum of the \(n\)-tuple \((A_1,\ldots,A_n)\) (if the Weyl calculus is defined which requires some exponential growth estimate). In Clifford analyis, a similar symmetric average as above occurs when the polynomial \(f\) is extended to a monogenic function which motivates some connections employed in the present monograph. After a thorough introduction to the Weyl calculus and Clifford analysis in Sections~1--3, Section~4 makes use of the analogue of the Cauchy integral formula in Clifford analysis to define and study a ``monogenic calculus'' as an analogue of the Riesz-Dunford calculus. It applies for an \(n\)-tuple of bounded operators for which \(\xi_1A_1+\cdots+\xi_nA_n\) has a real spectrum for real \(\xi_j\). This spectral condition is also necessary for the exponential growth estimate of the Weyl calculus, but it is in general not sufficient. If the growth estimate also holds, the Weyl calculus and the monogenic calculus coincide. Section~5 is devoted to the finite-dimensional settting. It is shown that the above spectral condition is for matrices even sufficient for the exponential growth estimate of the Weyl calculus. In this case, a formula, originally due to E. Nelson, allowing a more explicit calculation, is derived. Moreover, the joint spectrum …
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81S40
zbMATH DE Number: 2122040
zbMATH Keywords: noncommutative operator calculus
zbMATH Keywords: Weyl calculus
zbMATH Keywords: Clifford analysis
zbMATH Keywords: monogenic calculus
zbMATH Keywords: monogenic extension
zbMATH Keywords: noncommutative algebra
zbMATH Keywords: Feynman \(\mu\)-calculus
zbMATH Keywords: Feynman integral
author: Brian Jefferies
MaRDI profile type: Publication
full work available at URL: b97327
OpenAlex ID: W1695030651
DOI: 10.1007/B97327
description: scientific article; zbMATH DE number 2122040