On a Christoffel Transformation for Matrix Measures Supported on the Unit Circle

Abstract

Let 𝜎 be a Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of the orthogonal matrix polynomials associated with the Christoffel matrix transformation of 𝜎 defined by 𝑑𝜎𝑐𝑚(𝑧)=𝑊𝑚(𝑧)𝐻𝑑𝜎(𝑧)𝑊𝑚(𝑧), where 𝑊𝑚(𝑧)=∏𝑚𝑗=1(𝑧𝐈−𝐴𝑗) and 𝐴𝑗 is a square matrix for 𝑗=1,…,𝑚. Moreover, we study the relative asymptotics of the associated orthogonal matrix polynomials when 𝜎𝑐𝑚 satisfies a matrix condition in the diagonal case. Some illustrative examples are considered.