Christoffel transformation for a matrix of Bi-variate measures.

Abstract

We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms <center dot, center dot >((R) over cap) and <center dot, center dot >((L) over cap) < P(z(1)), Q(z(2))((R) over cap) = (TxT)integral P(z(1))dagger L(z(1))d mu(z(1), z(2)) Q(z(2)), P, Q is an element of L-pxp[z] < P(z(1)), Q(z(2))>(L) over cap = (TxT)integral P(z(1))L(z(1))d mu(z(1), z(2))Q(z(2)), where mu(z1, z2) is a matrix of bi-variate measures supported on T x T, with T the unit circle, L pxp[ z] is the set of matrix Laurent polynomials of size p x p and L(z) is a special polynomial in L pxp[ z]. A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to <center dot, center dot >((R) over cap) and resp <center dot, center dot >((L) over cap) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to d mu(z(1), z(2)) is given.