Circular operators on minimal norm ideals of B(H)

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. In this article the notion of circular operator is extended to the Banach space setting. In particular, this property is considered for elementary operators of lengths one and two acting on minimal norm ideals of B(H). Necessary and sufficient conditions for the circularity of generalized derivations and Lu¨ders operators are also obtained. 1. Introduction The notion of circular operators on a Hilbert space was introduced by Gellar in [4]. An operator L 2 B(H) is said to be circular if L is unitarily equivalent to L, for every of modulus 1. This idea was later pursued by Arveson et al. in [1] where the circularity of weighted translation operators on L 2 (X,) was studied. In this article, we extend this notion to operators on Banach spaces. Given a Banach space X and a complex number of modulus 1, we say that a bounded operator L on X is-circular if and only if L and L are isometrically equivalent. This means that there exists a surjective isometry S : X ! X such that S L S