Unbounded operators having self-adjoint or normal powers and some related results

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.