Spectral and pseudospectral functions of various dimensions for symmetric systems

Abstract

The main object of the paper is a symmetric system J ′ − B(t)y = ⋋∆(t)y defined on an interval Ι = [a, b) with the regular endpoint a. Let φ(⋅, λ) be a matrix solution φ(⋅, λ) of this system of an arbitrary dimension, and let $$ \left( V\kern0.5em f\right)(s)={\displaystyle \underset{I}{\int }{\varphi}^{\ast}\left( t, s\right)\varDelta (t) f(t) d t} $$ be the Fourier transform of the function f(⋅) ∈ L 2 (I). We define a pseudospectral function of the system as a matrix-valued distribution function σ(·) of the dimension n σ such that V is a partial isometry from $$ {L}_{\varDelta}^2(I)\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em {L}^2\left(\sigma; \kern0.5em {\mathbb{C}}^{n_{\sigma}}\right) $$ with minimally possible kernel. Moreover, we find the minimally possible value of n σ and parametrize all spectral and pseudospectral functions of every possible dimensions n σ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; Sakhnovich, Sakhnovich and Roitberg; Langer and Textorius.