Some Remarks on the Operator T^*T

Abstract

This note deals with the operator $T^{\ast }T$, where $T$ is a densely defined operator on a complex Hilbert space. We reprove a recent result of Z. Sebestyén and Zs. Tarcsay [13]: If $T^{\ast }T$ and $T{T}^{\ast }$ 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. Arlinskiĭ and Yu. Kovalev [1], [2].

6 pages