Local inverse spectral problems for rational matrix functions

Abstract

Given a regular rational matrix function A(z), we describe the set of vector-valued functions g(z) of the form g(z)=A(z)f(z) for some vector valued function f(z) which is analytic at the prescribed point λ. The description is in terms of a “canonical set of spectral data” for A at λ. We solve the inverse problem of characterizing when a given set of data is a canonical set of spectral data for some rational matrix function at each point in some subset of the complex plane. Our approach is to understand the connection between the rational nxn matrix function A and the shift invariant subspaces AH n 2 and AH n 2⊥ in L n 2 . As an application we show how the linear fractional map which parametrizes the set of all solutions of a matricial Nevanlinna-Pick interpolation problem can be found as the solution of a local inverse spectral problem.