  This note deals with the operator $T^*T$, where $T$ is a densely defined
operator on a complex Hilbert space. We reprove a recent result of Z.
Sebesty\'en and Zs. Tarcsay [13]: If $T^*T$ and $TT^*$ are self-adjoint, then
$T$ is closed. In addition, we describe the Friedrichs extension of $S^2$,
where $S$ is a symmetric operator, recovering results due to Yu. Arlinski\u{i}
and Yu. Kovalev [1], [2].
