A general factorization approach to the extension theory of nonnegative operators and relations

The Kreĭn-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 Kreĭn-von Neumann and the Friedrichs extensions and to the notion of positive closability (the Kreĭn-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.