The Closed Extensions of a Closed Operator

TL;DR

This paper characterizes all closed extensions of a densely defined closed operator on a Hilbert space, generalizing classical results and providing new constructions of operators with specific extension properties.

Contribution

It establishes a one-to-one correspondence between closed extensions and certain subspaces, extending classical theory to more general settings including Gel'fand triples.

Findings

Characterization of all closed extensions via subspaces of the adjoint domain.

Construction of a sequence of operators converging to a non-densely defined operator.

Examples of non-closable extensions and selfadjoint Laplacian extensions.

Abstract

Given a densely defined and closed 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. As applications we construct: (i) a sequence of densely defined operators that converge in the generalized sense to a non-densely defined operator, (ii) a non-closable extension of a symmetric operator and (iii) selfadjoint extensions of Laplacians with a generalized boundary condition.