Positive symmetric quotients and their selfadjoint extensions

We define a quotient $B/A$ of bounded operators $A$ and $B$ on a Hilbert space $H$ with a kernel condition $\ker A\subset \ker B$ as the mapping $Au\to Bu$, $u\in H$. A quotient $B/A$ is said to be positive symmetric if $A^*B=B^*A\ge 0$. In this paper, we give a simple construction of positive selfadjoint extensions of a given positive symmetric quotient $B/A$.