On the Friedrichs method of extension of a positive definite operator to a self-adjoint operator
Abstract
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.
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References
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