Spectral and pseudospectral functions of various dimensions for symmetric systems

TL;DR

This paper characterizes spectral and pseudospectral functions for symmetric systems of various dimensions, providing a parameterization using Nevanlinna boundary parameters and extending previous foundational results.

Contribution

It introduces a framework for defining and classifying spectral and pseudospectral functions of symmetric systems across different dimensions, including minimal dimension determination and parameterization.

Findings

Defined pseudospectral functions as matrix-valued distributions with minimal kernel

Determined the minimal possible dimension of pseudospectral functions

Parameterized all spectral and pseudospectral functions using Nevanlinna boundary parameters

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.