A class of linear kinetic equations in a Krein space setting (Q1110067)

This paper continues the series of applications of the spaces with an indefinite metric in transport theory, initiated by Ball and Greenberg in 1975. In particular it extends some parts of the more recent Ganchev's Ph. D. Thesis (1986). The authors study the following abstract transport equations \[ \frac{d}{dx}T\psi (x)=-A\psi (x),\quad x>0, \] which describe many important particular physical processes; here x is a spatial variable, \(\psi\) (x)\(\in H\), a Hilbert space, and A, T are linear operators on H. The boundary conditions concern the incoming flux \(Q_+\psi (0)=\phi_+\), where \(Q_+\) is the positive spectral projector for the self-adjoint and injective operator T, and the behavior at infinity. It is shown that in many cases the operator \(T^{-1}A\) generates a bisemigroup with separating projectors \(P_{\pm}\), hence every solution has the form \(\exp (-xT^{-1}A)h\) for some \(h\in P_+H\), and consequently the problem is uniquely solvable if and only if \(Q_+:P_+H\to Q_+H\) is a bijection. In order to include the weak solutions and the unbounded operators, the authors develop the analysis in general Krein spaces. Some of the results are applied to the Boltzmann-Fokker-Planck equation.