Spectral and pseudospectral functions of various dimensions for symmetric systems

The main object of the paper is a symmetric system $J y-B(t)y=lD(t) y$ 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 $(Vf)(s)=intlimits_cI f^*(t,s)D(t)f(t),dt$ 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.