Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2017; 54(1): 215-223

Online first article November 1, 2016      Printed January 31, 2017

https://doi.org/10.4134/BKMS.b160037

Copyright © The Korean Mathematical Society.

Bounded and unbounded operators similar to their adjoints

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

MSC numbers: Primary 47A62; Secondary 47A05, 47A12, 47B20, 47B25