Analytic functional calculus for two operators

TL;DR

This paper develops an analytic functional calculus for pairs of operators using integral representations, enabling new insights into differential equations, Sylvester equations, and operator calculus properties.

Contribution

It introduces a novel integral-based functional calculus for two operators, extending existing operator theory and applications.

Findings

Representation of impulse responses for second order differential equations

Solution formulas for Sylvester equations

Analysis of the differential of the functional calculus

Abstract

Properties of the mappings \begin{align*} C&\mapsto\frac1{(2\pi i)^2}\int_{\Gamma_1}\int_{\Gamma_2}f(\lambda,\mu)\,R_{1,\,\lambda}\,C\, R_{2,\,\mu}\,d\mu\,d\lambda, C&\mapsto\frac1{2\pi i}\int_{\Gamma}g(\lambda)R_{1,\,\lambda}\,C\, R_{2,\,\lambda}\,d\lambda \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.