Pick matrices for schur multipliers

Abstract

It is well known that the classical Nevanlinna-Pick problem for holomorphic contractive functions in the open unit disk is solvable if and only if a matrix P with entries of the form (Formula Presented) that is based on the data of the problem is positive semidefinite. The purpose of this purely expository note is to draw attention to another matrix that arises in the theory of interpolation problems for multipliers which deserves to be better known. This matrix and other more general forms are discussed in [AlB97] and [AlBL96]. Our interest in this problem was aroused by the formula (Formula Presented), that was mentioned by M.A. Kaashoek in a lecture at the IWOTA conference in Blacksburg, Virginia, as a byproduct of his joint investigations with C. Foias and A.E. Frazho [FFK02] into constrained lifting problems.