On Kreǐn's extension theory of nonnegative operators

Abstract

In M. G. Kreǐn's extension theory of nonnegative operators a complete description is given of all nonnegative selfadjoint extensions of a densely defined nonnegative operator. This theory, the refinements to the theory due to T. Ando and K. Nishio, and its extension to the case of nondensely defined nonnegative operators is being presented in a unified way, building on the completion of nonnegative operator blocks. The completion of nonnegative operator blocks gives rise to a description of all selfadjoint contractive extensions of a symmetric (nonselfadjoint) contraction. This in turn is equivalent to a description of all nonnegative selfadjoint relation (multivalued operator) extensions of a nonnegative relation. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.