Some Preorder on Operators in Semi-Hilbertian Spaces

Let A be a positive operator in B(H): Then for x; y 2 H; the semi- inner product .x; y.A = .Ax; y.; and the seminorm .x.A = .A 12 x. are de- ned on complex Hilbert space (H; . ; .): The aim of this work is to investigate a preorder on semi-Hilbertian space operators, it is called A-majorizarion. In some sense, the A-majorizarion is equivalent to Barnes s majorization. Some equivalent Theorems are obtained. The relations between A-majorization, range inclusion and A-numerical radius are studied.