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 $Q:\mathcal{D}(Q)\to \mathcal{L}(\mathcal{H})$ is investigated. We provide a characterization of pole cancellation functions $\mathit{\psi }(z)$ 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 $\mathit{\psi }(z)$. Moreover, it leads to an explicit formula for the associated functions $(z):=Q(z)\mathit{\psi }(z)$, which are root functions of order $k$ at the zero $\beta$ of $Q^{-1}$. The results are illustrated by an example.