A Study on the Joint Maximal Numerical Range of Aluthge Transform

Abstract

The Aluthge transform Te of a bounded linear operator T on a complex Hilbert space X is the operator |T| 1 2 U|T| 1 2 . Here, T = U|T| is any polar decomposition of T with U a partial isometry and |T| = (T ∗T) 1 2 . This study of the Aluthge transform Te was introduced and studied by Aluthge in his study of p-hyponormal operators. Since its conception, this notion has received much attention for a single operator T. In order to understand the joint behaviour of Aluthge transform of several operarors T1, ..., Tm, researchers such as Cyprian have studied the Aluthge transform of an m−tuple operator T = (T1, ..., Tm). For instance, the properties of the joint essential numerical range of Aluthge transform for an m−tuple operator T = (T1, ..., Tm) were studied by Cyprian. However, nothing is known about the joint maximal numerical range of Aluthge transform Te of an m−tuple operator T = (T1, ..., Tm). This paper focuses on the study of the properties of the joint maximal numerical range of Aluthge transform for an m−tuple operator T = (T1, ..., Tm). This study will help in the development of the research on hyponormal operators and semi-hyponormal operators.