Cyprian Omukhwaya Sakwa
Kibabii University 1st International Conference; June 22-24,2016
Let T = UT be any polar decomposition of an operator T with U a partial isometry. The Aluthge transform T = |T|½ U|T|½ was first introduced by A. Aluthge in 1990 in his study of p-hyponormal operators. Properties of the transform have since been investigated by several authors. Some elementary spectral and numerical range prop-erties and related results of Aluthge transform have been obtained. For instance, in 2002, Yuan Wu proved that the closure of the numerical range of Aluthge transform of the operator T is contained in that of T. In 2007, Guoxing Ji, Ni Liu and Ze Li together showed that the essential numerical range of Aluthge transform is contained in the essential numerical range of T: However, the properties of Aluthge transform have not been exhaustively studied. For instance, the joint essential maximal nu-merical range of Aluthge transform has not been studied. This paper will therefore introduce and study the joint essential maximal numerical range of T and establish its properties. In particular, this paper shows that the joint essential maximal nu-merical range of T is contained in the joint essential maximal numerical range of an m-tuple operator T = T1;….; Tm: Further, we introduce and study the notion of the joint maximal numerical range of T. The results of this study will be helpful in the development of the research on numerical ranges and may also be applied by mathematicians in solving several problems in operator theory.