ON THE FACTORIZATION OF MAXIMAL SECTORIAL EXTENSIONS SEPPO HASSI, ADRIAN SANDOVICI, HENK DE SNOO, AND HENRIK WINKLER 1. Abstract If S is a nonnegative linear relation in a Hilbert space H, then there are two extreme nonnegative selfadjoint extensions, namely the Kre˘ ın-von Neumann exten- sion S N and the Friedrichs extension S F , and all nonnegative selfadjoint extensions of S are between S N and S F . These extreme extensions and, in fact, all extremal nonnegative extensions (see [7]) can be factorized by means of the Kre˘ ın-von Neu- mann extension S N ; see [6, 8, 9]. The purpose of the present paper is to develop a similar theory for sectorial relations. The general theory of extensions of sectorial operators and relations is due to Yu.M. Arlinski˘ ı [3, 1, 2, 4, 5]. A linear relation S in a Hilbert space H is said to be sectorial with vertex at the origin and semi-angle α, α ∈ [0,π/2), if (1.1) (tan α)Re (f ′ ,f ) ≥|Im (f ′ ,f )|, {f,f ′ }∈ S. A linear relation S in a Hilbert space H is said to be maximal sectorial if the existence of a sectorial relation T in H with S ⊂ T implies S = T . Then there exist two maximal sectorial extensions of S in H, namely the Kre˘ ın-von Neumann extension S N and the Friedrichs extension S F , cf. [3]. It will be shown that the Kre˘ ın-von Neumann extension S N has a factorization S N = J ∗∗ (I + iB)J ∗ , where J is a relation from an auxiliary space H S to H, and B is a bounded selfadjoint operator in H S , and that the Friedrichs extension has a similar factorization S F = Q ∗ (I + iB)Q ∗∗ , where Q ∗∗ is a certain restriction of J ∗ . The factorizations of S N and S F in the general case provide a novel approach to notions such as disjointness, transversality, and equality of S N and S F ; and to the notion of positive closability of S (S N being an operator). A maximal sectorial extension H of S is said to be extremal if (1.2) inf { Re (f ′ - h ′ ,f - h): {h, h ′ }∈ S } =0 for all {f,f ′ }∈ H. This definition goes back to Yu.M. Arlinski˘ ı [3]. In the present paper the extremal extensions are characterized by factorizations in the general case. References [1] Yu.M. Arlinski˘ ı, ”Positive spaces of boundary values and sectorial extensions of nonnegative symmetric operators”, Ukrainian Math. J., 40 (1988), 8–15. [2] Yu.M. Arlinski˘ ı, ”Maximal sectorial extensions and closed form associated with them”, Ukrainian Math. J., 48 (1996), 723–739. 1991 Mathematics Subject Classification. Primary 47A06, 47B44; Secondary 47A12, 47B25. Key words and phrases. Sectorial relation, Hilbert space, Friedrichs extension, Kre˘ ın-von Neu- mann extension. 1