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Complex Analysis and Operator Theory

, Volume 13, Issue 8, pp 3979–4005 | Cite as

Christoffel Transformation for a Matrix of Bi-variate Measures

  • Juan C. García-ArdilaEmail author
  • Manuel Mañas
  • Francisco Marcellán
Article
  • 42 Downloads

Abstract

We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms \(\left\langle {\cdot ,\cdot }\right\rangle _{{{\hat{R}}}}\) and \(\left\langle {\cdot ,\cdot }\right\rangle _{{{\hat{L}}}}\)
$$\begin{aligned} \begin{array}{cc} \langle P(z_1),Q(z_2)\rangle _{{\hat{R}}}=\displaystyle \int \limits _{{\mathbb {T}}\times {\mathbb {T}}} P(z_1)^\dag L(z_1)d\mu (z_1,z_2) Q(z_2),&{}\\ &{}\quad P,Q\in {\mathbb {L}}^{p\times p}[z]\\ \left\langle {P(z_1),Q(z_2)}\right\rangle _{{\hat{L}}}=\displaystyle \int \limits _{{\mathbb {T}}\times {\mathbb {T}}} P(z_1)L(z_1)d\mu (z_1,z_2) Q(z_2)^{\dag },&{} \end{array} \end{aligned}$$
where \(\mu (z_1,z_2)\) is a matrix of bi-variate measures supported on \({\mathbb {T}}\times {\mathbb {T}},\) with \({\mathbb {T}}\) the unit circle, \(L^{p\times p}[z]\) is the set of matrix Laurent polynomials of size \(p\times p\) and L(z) is a special polynomial in \(L^{p\times p}[z]\). A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to \(\left\langle {\cdot , \cdot }\right\rangle _{{{\hat{R}}}}\), (resp. \(\left\langle {\cdot , \cdot }\right\rangle _{{{\hat{L}}}}\)) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to \(d\mu (z_1,z_2)\) is given.

Keywords

Matrix biorthogonal polynomials Matrix-valued measures Nondegenerate continuous bilinear forms Gauss–Borel factorization Matrix Christoffel transformations Quasideterminants Block CMV matrices 

Mathematics Subject Classification

42C05 15A23 30C10 

Notes

Acknowledgements

The authors thank the referees by the careful revision of the manuscript. Their suggestions and remarks have contributed to improve its presentation. The work of Juan C. García-Ardila and Francisco Marcellán has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía, Industria y Competitividad of Spain, Grant [MTM2015-65888-C4-2-P]. The work of Manuel Mañas has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía, Industria y Competitividad of Spain, Grant [MTM2015-65888-C4-3-P].

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Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada a la Ingeniería IndustrialUniversidad Politécnica de MadridMadridSpain
  2. 2.Departamento de Física TeóricaUniversidad Complutense de MadridMadridSpain
  3. 3.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

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