Abstract. The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space H, let H be an auxiliary Hilbert space, let
A nonnegative selfadjoint extension A of a nonnegative operator A is callednew construction of all extremal extensions of a nonnegative densely defined operator will be presented. It employs a fixed auxiliary Hilbert space to factorize each extremal extension. Various functional-analytic interpretations of extremal extensions are studied and some new types of characterizations are obtained. In particular, a purely analytic description of extremal extensions is established, based on a class of functions introduced by M. G. Krein and I. E. Ovcarenko.
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