On the Friedrichs method of expansion of a positively defined operator to a self-adjoint operator

Abstract

The Friedrichs' technique of extending a positively defined operator in a Hilbert space to a self-adjoint operator (and hence having everywhere a bounded inverse) is at present apparently the simplest way of proving existence theorems for solutions of boundary value problems for self-adjoint equations of elliptic type. Indeed, according to Friedrichs, the matter is reduced to the proof of an inequality expressing the positive definiteness of the operator in the corresponding Hilbert space, after which the existence of a generalized solution of the problem becomes obvious. At the same time, the very procedure of operator expansion, which has in each case its specific functional-theoretic content, indicates in what sense this generalized solution should be understood. The proposed note aims to show that Friedrichs' result is also valid for positively defined operators acting from one Banach space to another space conjugate to it. As an application, some results on the solvability of elliptic boundary value problems are given. A correlation integral equation is an equation of the following form (see article). The purpose of this paper is to investigate the system of fundamental functions of the equation. We will carry out the reasoning for the case of symmetric correlation (see article). However, this restriction can be easily removed and non-symmetric correlation can be considered if we pass to the system of integral equations using the Hilbert-Schmidt theory. Let us prove a few theorems (see article).