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 = .a12x. are de-ned on complex hilbert space (h; .
;
.): the aim of this work is to investigatea preorder on semi-hilbertian space operators, it is called a-majorizarion. insome sense, the a-majorizarion is equivalent to barnes's majorization. someequivalent theorems are obtained. the relations between a-majorization,range inclusion and a-numerical radius are studied.