A pole of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$m \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta \in \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> of a regular operator-valued function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q: \mathcal {D}(Q) \rightarrow \mathcal {L}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>:</mo> <mi mathvariant="script">D</mi> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is investigated. We provide a characterization of pole cancellation functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varvec{\psi }(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of Q(z) of order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k \le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation> at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> in terms of the coefficients of the Laurent expansion of Q. This characterization yields practical and explicit constructions of pole cancellation functions <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varvec{\psi }(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover, it leads to an explicit formula for the associated functions <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varvec{\hat{\varphi }}(z):= Q(z)\varvec{\psi }(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">φ</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which are root functions of order k at the zero <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$Q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. The results are illustrated by an example.

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

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2026年7月25日

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Muhamed Borogovac

摘要

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