Selfadjoint extensions of relations whose domain and range are orthogonal - Hassi, S.; Labrousse, J. Ph.; de Snoo, H. (Methods of Functional Analysis and Topology, 2020)
The selfadjoint extensions of a closed linear relation R from a Hilbert space H_1 to a Hilbert space H_2 are considered in the Hilbert space H_1H_2 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 H_1 H_2 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 Krein-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.