Some properties of the adjoint of the unbounded operators $T{T}^{\ast }$ and $T^{\ast }T$

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