IMPROVEMENTS OF A-NUMERICAL RADIUS FOR SEMI-HILBERTIAN SPACE OPERATORS

Let A be a bounded positive operator on a complex Hilbert space ( H , (center dot , center dot) ) . The semi -product ( x , y ) A : = ( Ax , y ) , x , y E H , induces a semi -norm II center dot II A on H . Let co A ( T ) and II T II A denote the A -numerical radius and the A -operator semi -norm of an operator T in semiHilbertian space ( H , (center dot , center dot) A ) , respectively. In this paper, some new bounds for the A -numerical radius of operators in semi -inner product space induced by A are derived. In particular, for T E B A ( H ) and a 0, we prove that co A 4 ( T ) 1 + 2 a 16 ( 1 +a ) II T A T + TT A II 2 A + 3 + 2 a 8 ( 1 +a) II T A T + TT A II A co A ( T 2 ) and co A 4 ( T ) s 1 + 2 a 8 ( 1 + a ) II T A T + TT A II 2 A + 1 2 ( 1 + a)co A 2 ( T 2 ) . It is worth noting that our results improve the existing A -numerical radius inequalities. Further, we also give a refinement inequality of A -operator semi -norm triangle inequality.