Spectral properties of noncommuting operators
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 is located within the joint numerical range in the case of two Hermitian matrices by geometric means; further cases like simultaneously triangularizable matrices are also discussed. In Section~6, unbounded operators are considered, and the above spectral condition is relaxed to some sectorial condition. To this end, some results about monogenic extension of functions in sectors are established. The final Section~7 observes a connection with the Feynman calculus which arose from the Feynman integral. The Feynman calculus generalizes the Weyl calculus in the sense that not necessarily the symmetric average plays the key role for polynomials but instead some ``disentangling'' operation determined by probability measures.
scientific article; zbMATH DE number 2122040