Schur Analysis for Matrices with a Finite Number of Negative Squares

Abstract

Writing the form associated with a positive matrix as a sum of positive squares, there appears a sequence of complex numbers determining the given matrix. In a certain sense, these calculations are equivalent to the classical algorithm of I.Schur [24] which associates a similar sequence of parameters to an analytic contractive function on the unit disc (such sort of parameters are known as Schur-Szego parameters). The formalism using these Schur-Szego parameters (we call it Schur analysis) can be extended to a general framework (operators on Hilbert spaces — see [5], where the operatorial version of the Schur-Szego parameters is called choice sequence) and can be used to solve some extension problems (as Caratheodory-Fejer problem, Nehari problem and so on — see [1], [5]) and to describe the Kolmogorov decomposition of positive-definite kernels on the set of integers — see [13].