J. OPERATOR THEORY 58:2(2007), 351–386 © Copyright by THETA, 2007 A GENERAL FACTORIZATION APPROACH TO THE EXTENSION THEORY OF NONNEGATIVE OPERATORS AND RELATIONS SEPPO HASSI, ADRIAN SANDOVICI, HENK DE SNOO, and HENRIK WINKLER Communicated by Florian-Horia Vasilescu ABSTRACT. The Kre˘ ın-von Neumann and the Friedrichs extensions of a non- negative linear operator or relation (i.e., a multivalued operator) are character- ized in terms of factorizations. These factorizations lead to a novel approach to the transversality and equality of the Kre˘ ın-von Neumann and the Friedrichs extensions and to the notion of positive closability (the Kre˘ ın-von Neumann extension being an operator). Furthermore, all extremal extensions of the non- negative operator or relation are characterized in terms of analogous factoriza- tions. This approach for the general case of nonnegative linear relations in a Hilbert space extends the applicability of such factorizations. In fact, the ex- tension 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. KEYWORDS: Nonnegative relation, Friedrichs extension, Kre˘ ın-von Neumann ex- tension, disjointness, transversality, positive closability, extremal extension. MSC (2000): Primary 47A06, 47A57, 47A63, 47B25; Secondary 47A07, 47B65. INTRODUCTION To illustrate the factorizations introduced in this paper consider the follow- ing simple completion problem. Let H be a Hilbert space with the orthogonal decomposition H = H 1 ⊕ H 2 . Let S 11 be a nonnegative bounded linear operator in H 1 , let S 21 be a bounded linear operator from H 1 to H 2 , and let S 12 = S ∗ 21 . The usual form of the completion problem requires to determine all bounded linear operators S 22 in H 2 , such that (0.1)  S 11 S 12 S 21 ∗  becomes a nonnegative bounded linear operator in H, cf. [7], [13], [16], [28], [29], [33], [35]. This completion problem has a solution if and only if ran S ∗ 21 ⊂ ran S 1/2 11 .