Functional calculus for noncommuting operators

Given a single bounded selfadjoint operator A acting on a Hilbert space, a function f (A) of A may be formed by the Riesz-Dunford functional calculus [GRAPHICS] as a contour integral about the spectrum sigma(A) of A if f is analytic in a neighbourhood of sigma(A), or, if f epsilon L-infinity (P-A) with respect to the spectral measure P-A of A, then f (A) may be represented by the Spectral Theorem for selfadjoint operators as [GRAPHICS] As is well known, both procedures give the same operator f (A) in the case that f is analytic in a neighbourhood of sigma(A). As mentioned in the beginning of Chapter 2, we are looking for a higher dimensional analogue of the Riesz-Dunford functional calculus for a system A of n bounded linear operators acting on a Banach space. In the noncommuting case when A is of Paley-Wiener type s, the Weyl calculus W-A considered in Chapter 1 plays the role of a spectral measure for a single selfadjoint operator, although W-A is generally an operator valued distribution of order greater than one. In this chapter, the Weyl calculus W-A (when it exists) is used to define the Cauchy kernel G(w)(A) for a higher dimensional analogue of the Riesz-Dunford functional calculus. As expected, the two calculi agree and the Cauchy kernel G(w)(A) is defined by alternative means when A fails growth estimates for exponentials.