Bounded and Unbounded Operators Similar to Their Adjoints

TL;DR

This paper investigates conditions under which operators similar to their adjoints are self-adjoint, extending classical results to both bounded and unbounded operators on Hilbert spaces.

Contribution

It provides new proofs and generalizations of classical theorems relating similarity to adjoints and self-adjointness for bounded and unbounded operators.

Findings

Unbounded closed operators similar to their adjoints via cramped unitaries are self-adjoint.

New proof of Berberian's bounded operator result.

Generalizations of Sheth and Williams' results on hyponormal unbounded operators.

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 famous results by Sheth and Williams.