Feynman's operational calculi for noncommuting operators: The monogenic calculus (Q1431046)

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.