Unbounded operators having self‐adjoint, subnormal, or hyponormal powers

We show that if a densely defined closable operator A is such that the resolvent set of A2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial p(A)$p(A)$. We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that Tn$T^n$ is normal for some n≥2$n\ge 2$, then T is normal. Hence a closed subnormal operator T such that Tn$T^n$ is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that Ap$A^p$ and Aq$A^q$ are self‐adjoint for some coprime numbers p and q, then A must be self‐adjoint.