# Journal of Operator Theory

Volume , Issue 2, Fall 2007 pp. 351-368.

A general factorization approach to the extension theory of nonnegative operators and relations**Authors**: Seppo Hassi (1), Adrian Sandovici (2), Henk de Snoo (3), and Henrik Winkler (4)

**Author institution:**(1) Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland

(2) Colegiul National ``Petru Rares'', 610101, Str. Stefan cel Mare, Nr. 4, Piatra Neam\c t, Romania

(3) Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, Nederland

(4) Institut fuer Mathematik, MA 6-4, Technische Universitaet Berlin, Strasse des 17. Juni 136, 10623 Berlin, Deutschland

**Summary:**The Krein-von Neumann and the Friedrichs extensions of a nonnegative linear operator or relation (i.e., a multivalued operator) are characterized in terms of factorizations. These factorizations lead to a novel approach to the transversality and equality of the Krein-von Neumann and the Friedrichs extensions and to the notion of positive closability (the Krein-von Neumann extension being an operator). Furthermore, all extremal extensions of the nonnegative operator or relation are characterized in terms of analogous factorizations. This approach for the general case of nonnegative linear relations in a Hilbert space extends the applicability of such factorizations. In fact, the extension theory of densely and nondensely defined nonnegative relations or operators fits in the same framework. In particular, all extremal extensions of a bounded nonnegative operator are characterized.

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