The Closed Extensions of a Closed Operator

Abstract

Given a densely defined and closed (but not necessarily symmetric) operator A acting on a complex Hilbert space \({\mathcal {H}}\), we establish a one-to-one correspondence between its closed extensions and subspaces \({\mathfrak {M}}\subset {\mathcal {D}}(A^*)\), that are closed with respect to the graph norm of \(A^*\) and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of \(A^*\). After this, we will express our results using the language of Gel’fand triples generalizing the well-known results for the selfadjoint case.