LAUSR

AbstractWe consider the first-order symmetric system Jy′ − A(t)y = λΔ(t)y with n × n-matrix coefficients defined on an interval [a; b) with the regular endpoint a. It is assumed that the deficiency indices N± of the system satisfy the equality N_ ≤ N+ = n. The main result is the parametrization of all pseudospectral functions σ(·) of any possible dimension n????≤ n in terms of a Nevanlinna parameter τ = {C0(λ),  C1(λ)}. Such parametrization is given by the linear-fractional transform mτλ=C0λw11λ+C1λw21λ−1C0λw12λ+C1λw22λ$$ {m}_{\tau}\left(\uplambda \right)={\left({C}_0\left(\uplambda \right){w}_{11}\left(\uplambda \right)+{C}_1\left(\uplambda \right){w}_{21}\left(\uplambda \right)\right)}^{-1}\left({C}_0\left(\uplambda \right){w}_{12}\left(\uplambda \right)+{C}_1\left(\uplambda \right){w}_{22}\left(\uplambda \right)\right) $$ and the Stieltjes inversion formula for m???? (λ). We show that the matrix Wλ=wijλi,j=12$$ W\left(\uplambda \right)={\left({w}_{ij}\left(\uplambda \right)\right)}_{i,j=1}^2 $$ has the properties similar to those of the resolvent matrix in the extension theory of symmetric operators. The obtained results develop the results by A. Sakhnovich; Arov and Dym; and Langer and Textorius.