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

We study general (not necessarily Hamiltonian) first-order symmetric system $J{y}^{\mathrm{\prime }}-B(t)y=\text{D}(t)f(t)$ on an interval $\text{cI}=[a,b)$ with the regular endpoint $a$ and singular endpoint $b$. It is assumed that the deficiency indices $n_{\pm }(\text{Tmi})$ of the corresponding minimal relation $\text{Tmi}$ in $\text{LI}$ satisfy $n_{-}(\text{Tmi})\le n_{+}(\text{Tmi})$. We describe all generalized resolvents $y=R(\l)f, ; f\in\LI,$of$\Tmi$intermsofboundaryproblemswith$\l$-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 $\text{Om}(\text{l})$ of the system immediately in terms of boundary conditions. Such a parametrization is given both by the block representation of $\text{Om}(\text{l})$ 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.