Complex Analysis and Operator Theory

, Volume 13, Issue 8, pp 3979–4005

# Christoffel Transformation for a Matrix of Bi-variate Measures

• Juan C. García-Ardila
• Manuel Mañas
• Francisco Marcellán
Article

## 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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