Unbounded operators having self‐adjoint, subnormal, or hyponormal powers
Souheyb Dehimi, Mohammed Hichem Mortad
Abstract
Abstract We show that if a densely defined closable operator A is such that the resolvent set of A 2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial . 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 is normal for some , then T is normal. Hence a closed subnormal operator T such that is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that and are self‐adjoint for some coprime numbers p and q , then A must be self‐adjoint.