Null-Pole Subspace for Matrix Functions with J-Unitary Values on the Imaginary Axis or Unit Circle

Abstract

In this chapter we study rational matrix functions which have J-unitary values on the imaginary line or the unit circle, where J is a fixed signature matrix (recall that a square matrix J is called a signature matrix if J 2 = I, J = J*). A basic tool for the study of such functions is the Redheffer transform which gives a connection between J-unitary and unitary valued functions. The main goal is to establish the connection between structural properties of the associated null-pole subspace and the equivalence of certain analyticity properties with the number of negative squares of an associated kernel function (Theorems 13.2.3 and 13.2.4); these results are needed in Part V where J-unitary valued functions are used to give linear fractional maps which parametrize all solutions of an interpolation problem.