On the adjoint of Hilbert space operators

TL;DR

This paper provides necessary and sufficient conditions for an operator to be the adjoint of an unbounded operator in Hilbert spaces, with implications for characterizing various classes of operators.

Contribution

It introduces a new operator matrix approach to characterize adjoints and related properties of unbounded operators in Hilbert spaces.

Findings

Characterization of when an operator is the adjoint of another.

Conditions for operators to be closed, normal, or selfadjoint.

Proof that $T^*T$ always has a positive selfadjoint extension.

Abstract

In general, it is a non trivial task to determine the adjoint $S^*$ of an unbounded operator $S$ acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator $T$ to be identical with $S^*$. In our considerations, a central role is played by the operator matrix $M_{S,T}=\left(\begin{array}{cc} I & -T\\ S & I\end{array}\right)$. Our approach has several consequences such as characterizations of closed, normal, skew- and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that $T^*T$ always has a positive selfadjoint extension.