On the adjoint of Hilbert space operators

Abstract

In general, it is a non trivial task to determine the adjoint $S^{\ast }$ 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 $S^{\ast }$. 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 $T^{\ast }T$ always has a positive selfadjoint extension.

19 pages