Adjoint Pairs and Unbounded Normal Operators

TL;DR

This paper studies adjoint pairs of unbounded operators on Hilbert spaces, characterizing their extensions and normal operators, especially when one operator extends to a bounded invertible normal operator, with implications for boundary triplets.

Contribution

It introduces a boundary triplet framework for adjoint pairs with regular points at zero and characterizes their proper extensions as normal operators, including special cases with bounded inverses.

Findings

Proper extensions correspond to closed subspaces of defect spaces.

Normal operators are characterized when $B$ is formally normal with equal domains.

Descriptions of normal extensions when $B$ has a bounded inverse and defect space dimension is one.

Abstract

An adjoint pair is a pair of densely defined linear operators $A, B$ on a Hilbert space such that $\langle Ax,y\rangle=\langle x,By\rangle$ for $x\in \cD(A), y \in \cD(B).$ We consider adjoint pairs for which $0$ is a regular point for both operators and associate a boundary triplet to such an adjoint pair. Proper extensions of the operator $B$ are in one-to-one correspondence $T_\cC\leftrightarrow \cC $ to closed subspaces $\cC$ of $\cN(A^*)\oplus\cN(B^*)$. In the case when $B$ is formally normal and $\cD(A)=\cD(B)$, the normal operators $T_\cC$ are characterized. Next we assume that $B$ has an extension to a normal operator with bounded inverse. Then the normal operators $T_\cC$ are described and the case when $\cN(A^*)$ has dimension one is treated.