Spectral and pseudospectral functions of various dimensions for symmetric systems

Abstract

The main object of the paper is a symmetric system $J y'-B(t)y=\l\D(t) y$ defined on an interval $\cI=[a,b) $ with the regular endpoint $a$. Let $\f(\cd,\l)$ be a matrix solution of this system of an arbitrary dimension and let $(Vf)(s)=\int\limits_\cI \f^*(t,s)\D(t)f(t)\,dt$ be the Fourier transform of the function $f(\cd)\in L_\D^2(\cI)$. We define a pseudospectral function of the system as a matrix-valued distribution function $\s(\cd)$ of the dimension $n_\s$ such that $V$ is a partial isometry from $L_\D^2(\cI)$ to $L^2(\s;\bC^{n_\s})$ with the minimally possible kernel. Moreover, we find the minimally possible value of $n_\s$ and parameterize all spectral and pseudospectral functions of every possible dimensions $n_\s$ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; A.~Sakhnovich, L.~Sakhnovich and Roitberg; Langer and Textorius.