  We show that a densely defined closable operator $A$ such that the resolvent
set of $A^2$ is not empty is necessarily closed. This result is then extended
to the case of a polynomial $p(A)$. We also generalize a recent result by
Sebesty\'en-Tarcsay concerning the converse of a result by J. von Neumann.
Other interesting consequences are also given, one of them being a proof that
if $T$ is a quasinormal (unbounded) operator such that $T^n$ is normal for some
$n\geq2$, then $T$ is normal. By a recent result by Pietrzycki-Stochel, we
infer that a closed subnormal operator such that $T^n$ is normal, must be
normal.
  Another remarkable result is the fact that a hyponormal operator $A$, bounded
or not, such that $A^p$ and $A^q$ are self-adjoint for some co-prime numbers
$p$ and $q$, is self-adjoint. It is also shown that an invertible operator
(bounded or not) $A$ for which $A^p$ and $A^q$ are normal for some co-prime
numbers $p$ and $q$, is normal. These two results are shown using B\'{e}zout's
theorem in arithmetic.
