First-Order Operator Equations

Abstract

This chapter is, in a definite sense, central to our entire exposition. Here, by using the simplest entities, operator equations of the first order, generated by special classes of boundary value problems for partial differential equations that originate from the constructions and results presented above, we consider the whole circle of ideas that are of interest. In passing to more complicated entities the study of the corresponding questions turns out to involve a whole series of difficulties of a technical nature, whereas the constructions in this chapter are as transparent and elementary as possible. Let t∈ (0, b) =V _t , ℍ _t ≡ℍ _t (V _t ) the corresponding Hilbert space, and D _t : ℍ_t→ℍ_t, any of the operators generated by the Operation Dt. Then by an operator equation (or differential-operator equation) of the first order we ordinarily understand an equation of the form

Lu ≡ (A₀D_t+ A₁)u=f