On generalized resolvents and characteristic matrices of first-order symmetric systems

Abstract

We study general (not necessarily Hamiltonian) first-order symmetric system $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b) $ with the regular endpoint $a$ and singular endpoint $b$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the corresponding minimal relation $\Tmi$ in $\LI$ satisfy $n_-(\Tmi)\leq n_+(\Tmi)$. We describe all generalized resolvents $y=R(\l)f, \; f\in\LI,$ of $\Tmi$ in terms of boundary problems with $\l$-depending boundary conditions imposed on regular and singular boundary values of a function $y$ at the endpoints $a$ and $b$ respectively. We also parametrize all characteristic matrices $\Om(\l)$ of the system immediately in terms of boundary conditions. Such a parametrization is given both by the block representation of $\Om(\l)$ and by the formula similar to the well-known Krein formula for resolvents. These results develop the \u{S}traus' results on generalized resolvents and characteristic matrices of differential operators.