EXTENDED SPECTRUM, NUMERICAL RANGE OF OPERATORS AND ALUTHGE TRANSFORMATION AND INTERACTIONS

Abstract

around two themes : a transformation of B (H) introduced by Aluthge, the first part is the subject of the study of the transformation of Aluthge which has had a significant impact in recent years in operator theory . Optimal results on the stability of a certain number of operator classes, such as the class of partial isometrics and the classes associated with the asymptotic behavior of an operator, are provided. We also study the evolution of operator invariants, such as the minimum polynomial, the minimum function, the ascent and the descent, under the action of the transformation ; we compare more precisely the sequences of nuclei and images relating to the iterates of an operator and its Aluthge transform, and the second part will be devoted to the study of numerical range and a fairly recent spectral concept, called the extended spectrum operators...