Construction of pole cancellation functions at ordinary poles of operator-valued functions

A pole of order $$m \in \mathbb {N}$$ at $$\beta \in \mathbb {C}$$ of a regular operator-valued function $$Q: \mathcal {D}(Q) \rightarrow \mathcal {L}(\mathcal {H})$$ is investigated. We provide a characterization of pole cancellation functions $$\varvec{\psi }(z)$$ of Q(z) of order $$k \le m$$ at $$\beta $$ in terms of the coefficients of the Laurent expansion of Q. This characterization yields practical and explicit constructions of pole cancellation functions $$\varvec{\psi }(z)$$ . Moreover, it leads to an explicit formula for the associated functions $$\varvec{\hat{\varphi }}(z):= Q(z)\varvec{\psi }(z)$$ , which are root functions of order k at the zero $$\beta $$ of $$Q^{-1}$$ . The results are illustrated by an example.