Selfadjoint extensions of relations whose domain and range are orthogonal

The selfadjoint extensions of a closed linear relation R from a Hilbert space H1 to a Hilbert space H2 are considered in the Hilbert space H1⊕H2 that contains the graph of R. They will be described by 2 × 2 blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation S in H1⊕H2 that is induced by R. Such a relation is characterized by the orthogonality property dom S ⊥ ran S and it is nonnegative. All nonnegative selfadjoint extensions A, in particular the Friedrichs and Kreĭn-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that A belongs to the class of extremal extensions of S if and only if dom A ⊥ ran A. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of S and semibounded parameters describing them if and only if the operator part of R is bounded.