On generalized rezolvents and characteristic matrices of first-order symmetric systems - Mogilevskii, V. I. (Methods of Functional Analysis and Topology, 2014)
We study general (not necessarily Hamiltonian) first-ordersymmetric system J y’-B(t)y=Δ(t) f(t) on an interval I=[a,b) with the regular endpoint a and singular endpoint b. It isassumed that the deficiency indices n_±(T_) of thecorresponding minimal relation T_ in L_Δ^2(I) satisfyn_-(T_)≤ n_+(T_). We describe all generalized resolventsy=R(λ)f, fL_Δ^2(I), of T_ in terms of boundary problemswith λ-depending boundary conditions imposed on regular andsingular boundary values of a function y at the endpoints aand b respectively. We also parametrize all characteristicmatrices Ω(λ) of the system immediately in terms of boundaryconditions. Such a parametrization is given both by the blockrepresentation of Ω(λ) and by the formula similar to thewell-known Krein formula for resolvents. These results develop the Straus’ results on generalized resolvents and characteristicmatrices of differential operators.