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

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 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 𝑇 𝑛 is normal for some 𝑛 ≥ 2, then 𝑇 is normal. Hence a closed subnormal operator 𝑇 such that 𝑇 𝑛 is normal is itself normal.We also show that if a hyponormal (nonnecessarily bounded) operator 𝐴 is such that 𝐴 𝑝 and 𝐴 𝑞 are self-adjoint for some coprime numbers 𝑝 and 𝑞, then 𝐴 must be self-adjoint.