<p>In this paper, we study the structure of the differential operator algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \mathcal {D}(W) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">D</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and its associated eigenvalue algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \Lambda (W) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for matrix-valued orthogonal polynomials. While <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \Lambda (W) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \mathcal {D}(W) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">D</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, its simpler framework allows us to efficiently derive strong results about <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \mathcal {D}(W) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">D</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and its center <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \mathcal {Z}(W) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">Z</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We analyze the behavior of the center under Darboux transformations, establishing explicit relationships between the centers of Darboux-equivalent weights. These results are illustrated through the study of both reducible and irreducible matrix weights, including a detailed analysis of an irreducible Jacobi-type weight.</p>

Structure of operator algebras for matrix orthogonal polynomials

期刊论文
发表日期：

2025年8月8日

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Ignacio Bono Parisi,
Ines Pacharoni

摘要

In this paper, we study the structure of the differential operator algebra $ \mathcal {D}(W) $ $D (W)$ and its associated eigenvalue algebra $ \Lambda (W) $ $Λ (W)$ for matrix-valued orthogonal polynomials. While $ \Lambda (W) $ $Λ (W)$ is isomorphic to $ \mathcal {D}(W) $ $D (W)$ , its simpler framework allows us to efficiently derive strong results about $ \mathcal {D}(W) $ $D (W)$ and its center $ \mathcal {Z}(W) $ $Z (W)$ . We analyze the behavior of the center under Darboux transformations, establishing explicit relationships between the centers of Darboux-equivalent weights. These results are illustrated through the study of both reducible and irreducible matrix weights, including a detailed analysis of an irreducible Jacobi-type weight.