Selfadjoint extensions of relations whose domain and range are orthogonal

Abstract

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in ${\mathfrak H}_1 \oplus {\mathfrak H}_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative selfadjoint extensions $A$, in particular the Friedrichs and Kre\u{\i}n-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 ${\rm dom\,} A \perp {\rm ran\,} A$. 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.