Equations with Several Generalized Involutive Operators. Matrix Abstract Approach and Applications

Abstract

In Sections 14 and 19 we gave an approach to investigate “two-term” equations of the form Kϕ = (A + QB)ϕ = f with a generalized involutive operator Q. In this chapter we consider more general operators $K\phi= (A_1+ QA_2+\cdots+ Q^{n - 1} A_n )\phi= f$ and now the operators A _j and Q do not necessarily quasicommute as in Section 19. The investigations in Sections 14,19 and 20 were based on a simple regularization of the operators K and it was possible to carry out these investigations within the framework of scalar equations, without passage to systems of equations. In the case of more general equations of the above form, the passage to systems is necessary in a sense, at least without additional assumptions on quasicommutation of operators A _j with the operator Q.