  In this paper, we are concerned with conditions under which
$[p(T)]^*=\bar{p}(T^*)$, where $p(z)$ is a one-variable complex polynomial, and
$T$ is an unbounded, densely defined, and linear operator. Then, we deal with
the validity of the identities $\sigma(AB)=\sigma(BA)$, where $A$ and $B$ are
two unbounded operators. The equations $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$,
where $T$ is a densely defined closable operator, are also studied. A
particular interest will be paid to the equation $T^*T=p(T)$ and its variants.
Then, we have certain results concerning $nth$ roots of classes of normal and
nonnormal (unbounded) operators. Some further consequences and counterexamples
accompany our results.
