  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.
