On the class of extremal extensions of a nonnegative operator

A nonnegative selfadjoint extension Ãof a nonnegative operator A is called extremal if inf {(Ã)(ϕ) - f),ϕ - f) : ∈ dom A} = 0 for all ϕ ∈ dom Ã.A new 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.Ovearenko.