On Kreĭn’s extension theory of nonnegative operators. (English) Zbl 1076.47008

The paper under review contains a unified approach to Kreĭn’s extension theory of nonnegative operators (cf. M.G. Kreĭn [Mat. Sb., N. Ser. 20(62), 431–495 (1947; Zbl 0029.14103)]). Several generalizations, extensions and completions are also considered. Certain incomplete block operators are completed to nonnegative operators. The connection with Kreĭn’s shorted operators is discussed. The completion problem is equivalent to the description of all bounded selfadjoint extensions of a bounded symmetric operator. In particular, the set of selfadjoint contractive extensions of a given symmetric contraction is described as a certain closed interval. The extreme cases are characterized, leading to the Kreĭn’s uniqueness criterion for such a selfadjoint contractive extension. Several results concerning the theory of nonnegative selfadjoint extensions of a nonnegative relation, including complete characterizations of the Friedrichs and Kreĭn-von Neumann extensions (which are the extreme cases in this context), conclude the paper.


47A57 Linear operator methods in interpolation, moment and extension problems
47B25 Linear symmetric and selfadjoint operators (unbounded)
47A55 Perturbation theory of linear operators
47B65 Positive linear operators and order-bounded operators


Zbl 0029.14103
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