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

TL;DR

This paper investigates properties of unbounded operators with self-adjoint or normal powers, establishing new conditions under which such operators are normal or self-adjoint, with implications for operator theory.

Contribution

It extends existing results by providing new criteria for operators to be normal or self-adjoint based on their powers, using algebraic and spectral methods.

Findings

A densely defined closable operator with non-empty resolvent set of its square is closed.

If a quasinormal operator's power is normal, then the operator itself is normal.

Operators with certain powers being self-adjoint or normal under coprimality conditions are shown to be self-adjoint or normal.

Abstract

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.