Pseudospectral functions of various dimensions for symmetric systems with the maximal deficiency index

Resumo

We 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

\( {m}_{\tau}\left(\uplambda \right)={\left({C}_0\left(\uplambda \right){w}_{11}\left(\uplambda \right)+{C}_1\left(\uplambda \right){w}_{21}\left(\uplambda \right)\right)}^{-1}\left({C}_0\left(\uplambda \right){w}_{12}\left(\uplambda \right)+{C}_1\left(\uplambda \right){w}_{22}\left(\uplambda \right)\right) \)

and the Stieltjes inversion formula for m_???? (λ). We show that the matrix \( W\left(\uplambda \right)={\left({w}_{ij}\left(\uplambda \right)\right)}_{i,j=1}^2 \) 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.