Abstract. Let $A$ and $B$ be bounded operators defined on a Hilbert space $H$ with a kernel condition $\ker A \subset\ker B$. We define a quotient $B/A$ to be a mapping $Au \to Bu, u \in H$. It is known that the family of all quotients contains all closed operators. In this paper we investigate some properties of the smallest and the largest positive selfadjoint extensions of a given positive symmetric quotient $B/A$, that is, $A^{\ast }B=B^{\ast }A \ge0$ (with respect to an order introduced below).

AMS Subject Classification (1991): 47A05, 47A99

Received March 26, 1998 and in final form December 15, 1998. (Registered under 2682/2009.)