Souheyb Dehimi and Mohammed Hichem Mortad Ahmed Ben Bella, B.P. 1524, ElMenouar, Ahmed Ben Bella, B.P. 1524, El Menouar
Abstract : In this paper, we establish results about operators similar to their adjoints. This is carried out in the setting of bounded and also unbounded operators on a Hilbert space. Among the results, we prove that an unbounded closed operator similar to its adjoint, via a cramped unitary operator, is self-adjoint. The proof of this result works also as a new proof of the celebrated result by Berberian on the same problem in the bounded case. Other results on similarity of hyponormal unbounded operators and their self-adjointness are also given, generalizing well known results by Sheth and Williams.
Keywords : similarity, bounded and unbounded operators, closed, self-adjoint, normal, hyponormal operators, unitary cramped operators, numerical range