Analytic functional calculus for two operators

Abstract

Properties of the mappings \begin{align*} C&\mapsto\frac1{(2πi)^2}\int_{Γ_1}\int_{Γ_2}f(λ,μ)\,R_{1,\,λ}\,C\, R_{2,\,μ}\,dμ\,dλ, C&\mapsto\frac1{2πi}\int_Γg(λ)R_{1,\,λ}\,C\, R_{2,\,λ}\,dλ\end{align*} are discussed; here $R_{1,\,(\cdot)}$ and $R_{2,\,(\cdot)}$ are pseudo-resolvents, i.~e., resolvents of bounded, unbounded, or multivalued linear operators, and $f$ and $g$ are analytic functions. Several applications are considered: a representation of the impulse response of a second order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and an exploration of properties of the differential of the ordinary functional calculus.