Some Remarks on the Operator T^*T

TL;DR

This paper investigates properties of the operator T^*T in Hilbert spaces, reestablishes conditions for T to be closed, and describes the Friedrichs extension of S^2 for symmetric operators, building on recent and classical results.

Contribution

It provides a new proof of a recent result on self-adjointness of T^*T and TT^*, and characterizes the Friedrichs extension of the square of a symmetric operator.

Findings

If T^*T and TT^* are self-adjoint, then T is closed.

The Friedrichs extension of S^2 is explicitly described.

Connections to previous results by Arlinskind Kovalev are clarified.

Abstract

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].