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

A pole of order m ∈ N at β ∈ C of a regular operator-valued function Q: D(Q) → L(H) is investigated. We provide a characterization of pole cancellation functions ψ (z) of Q(z) of order k ≤ m at β in terms of the coefficients of the Laurent expansion of Q. This characterization yields practical and explicit constructions of pole cancellation functions ψ (z) . Moreover, it leads to an explicit formula for the associated functions φ ̂(z):= Q(z)ψ (z) , which are root functions of order k at the zero β of Q⁻¹ . The results are illustrated by an example.