On the class of extremal extensions of a nonnegative operator

A nonnegative selfadjoint extension A of a nonnegative operator A is called extremal if inf{(A(phi-f), phi-f) : f E dom A} = 0 for all W E dom (A) over tilde. 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. Ovcarenko.