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.

arXiv comment49 pages

References in corpus (2)

• Decay bounds for functions of matrices with banded or Kronecker structure
• Approximation of functions of large matrices with Kronecker structure

Cited by in corpus (3)

• Calculating a function of a matrix with a real spectrum
• An estimate of Green's function of the problem of bounded solutions in the case of a triangular coefficient
• Green's function of the problem of bounded solutions in the case of a block triangular coefficient