Convolution equations and the transport equation

In this chapter the factorization theory developed in the previous chapters is applied to solve a linear transport equation. It is known that the transport equation may be transformed into a Wiener-Hopf integral equation with an operator-valued kernel function (see [40]). An equation of the latter type can be solved explicitly if a canonical factorization of its symbol is available (cf., Sections 1.1 and 3.2). In our case the symbol may be represented as a transfer function, and to make the factorization the general factorization theorem of the second chapter can be applied. This requires that one finds an appropriate pair of invariant subspaces. In the case of the transport equation the choice of the subspaces is evident, but to prove that their direct sum is the whole space takes some effort. The latter is related to a new difficulty that appears here. Namely, in this case the curve cuts through the spectra of the main operator and the associate main operator. Nevertheless, due to the special structure of the operators involved, the factorization can be made and explicit formulas are obtained.