Friedrichs' technique of extending a positive definite 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 extension, 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 positive definite 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.

L. V. Kantorovich proposed a new direct method for solving problems of the calculus of variations, which he called the steepest descent method. As L. V. Kantorovich showed, the method can be successfully used for the approximate solution of linear functional equations in a Hilbert space. In addition, L. V. Kantorovich applied the steepest descent method to the problem of eigenvalues ​​of completely continuous operators. In application to this problem, the method consists of the following. In this note, a proof of the convergence of the process is given without the assumption of closeness of x0 to x*. In addition, a comparison of the steepest descent method with the known iterative method for calculating eigenvalues ​​is made, and a multi-step version of the method under consideration is investigated. In what follows, the assumption that m = 0 is not used.