An Overview of Functional Calculi for Noncommuting Operators

When X is a Banach space, the Riesz-Dunford functional calculus provides a mechanism for defining f(T) when T ∈ L (X). We consider the situation when A = (A1,..., An) ∈∏n i=1 L (X). In particular, we do not assume that the operators Ai commute. We give a brief survey of functional calculi over suitable classes of functions containing the polynomials. Ini-tially, we consider the Symmetric operator calculus defined over the over the polynomials, before examining the Weyl functional calculus over smooth functions when A satisfies an appropriate growth condition. Finally we employ Clifford algebras as a means to obtain the Monogenic functional calculus which is consistent with both the Weyl and Symmetric operator calculus