Unbounded operators having self-adjoint,subnormal, or hyponormal powers |
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Authors: | Souheyb Dehimi Mohammed Hichem Mortad |
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Institution: | 1. Department of Mathematics, Faculty of Mathematics and Informatics, University of Mohamed El Bachir El Ibrahimi, Bordj Bou Arréridj, El-Anasser, Algeria;2. Laboratoire d'analyse mathématique et applications. Département de Mathématiques, Ahmed Ben Bella, El Menouar, Oran, Algeria |
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Abstract: | 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 . 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. |
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Keywords: | Bézout's theorem in arithmetic closed operators hyponormal operators normal operators paranormal operators powers of operators quasinormal operators relatively prime numbers self-adjoint operators spectrum and resolvent set square roots subnormal operators unbounded operators |
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