The Monogenic Functional Calculus

The monogenic functional calculus Monogenic function calculus is a means of constructing functions of a finite system of bounded or unbounded operators. For bounded noncommuting operators, a polynomial function produces a polynomial of the operators in which all possible operator orderings are equally weighted. For example, for two bounded selfadjoint operators A 1, A 2, the operator 1 2 ( A 1 A 2 + A 2 A 1 ) $$\frac{1} {2}(A_{1}A_{2} + A_{2}A_{1})$$ is associated with polynomial z ↦ z 1 z 2 $$z\mapsto z_{1}z_{2}$$ in two variables by the monogenic functional calculus. The same formula applies just when the spectrum σ ( ξ 1 A 1 + ξ 2 A 2 ) $$\sigma (\xi _{1}A_{1} +\xi _{2}A_{2})$$ of a finite linear combination ξ 1 A 1 + ξ 2 A 2 $$\xi _{1}A_{1} +\xi _{2}A_{2}$$ of A 1 and A 2 is a subset of the real numbers for any ξ 1 , ξ 2 ∈ ℝ $$\xi _{1},\xi _{2} \in \mathbb{R}$$ .The article begins with a discussion of Clifford algebras and Clifford analysis and points out the connection with Weyl’s functional calculus for a finite system of selfadjoint operators in Hilbert space. The construction of the Cauchy kernel in the monogenic functional calculus is achieved with the plane wave decomposition of the Cauchy kernel in Clifford analysis. The formulation applies to any finite system ( A 1 , … , A n ) $$(A_{1},\ldots,A_{n})$$ of closed unbounded operators such that the spectrum σ ξ 1 A 1 + … + ξ n A n $$\sigma \left (\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\right )$$ of the closed operator ξ 1 A 1 + … + ξ n A n $$\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}$$ is contained in a two-sided sector in ℂ $$\mathbb{C}$$ for almost all ξ ∈ ℝ n $$\xi \in \mathbb{R}^{n}$$ . The connection with harmonic analysis and irregular boundary value problems is emphasized. The monogenic functional calculus may be applied to the finite commuting system D Σ = i ( D 1 , … , D n ) $$D_{\Sigma } = i(D_{1},\ldots,D_{n})$$ of differentiating operators on a Lipschitz surface Σ $$\Sigma $$ so that an H ∞ -functional calculus f ↦ f ( D Σ ) $$f\mapsto f(D_{\Sigma })$$ is obtained for functions f uniformly bounded and left monogenic in a sector containing almost all tangent planes to Σ $$\Sigma $$ .