Extremal extensions for the sum of nonnegative selfadjoint relations

The sum $A+B$ of two nonnegative selfadjoint relations (multi-valued operators) $A$ and $B$ is a nonnegative relation. The class of all extremal extensions of the sum $A+B$ is characterized as products of relations via an auxiliary Hilbert space associated with $A$ and $B$. The so-called form sum extension of $A+B$ is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with $A$ and $B$. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on $A$ and $B$.