A symmetric functional calculus for noncommuting systems of sectorial operators

Abstract

Given a system $A = (A_1, . . . , A_n)$ of linear operators whose real linear combinations have spectra contained in a fixed sector in $\mathbbC$ and satisfy resolvent bounds there, functions $f(A)$ of the system $A$ of operators can be formed for monogenic functions f having decay at zero and infinity in a corresponding sector in $\mathbb^n+1$. The paper discusses how the functional calculus $f \mapsto f(A)$ might be extended to a larger class of monogenic functions and its relationship with a functional calculus for analytic functions in a sector of $\mathbbC_n$.