Some properties of the adjoint of the unbounded operators $TT^*$ and $T^*T$

TL;DR

This paper examines the properties of adjoint operators for unbounded operators, specifically verifying certain identities involving $(TT^*)^*$ and $(T^*T)^*$ for densely defined operators.

Contribution

It provides a detailed analysis of the conditions under which the identities $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$ hold for unbounded operators.

Findings

Confirmed the identities hold for densely defined closable operators.

Clarified the conditions necessary for the identities to be valid.

Enhanced understanding of adjoint properties in unbounded operator theory.

Abstract

In this note, we mainly investigate the validity of the identities $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable (or symmetric) operator.