FURTHER IMPROVEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES FOR SEMI-HILBERTIAN SPACE OPERATORS

摘要

Several new improvements of the A-numerical radius inequalities for operators acting on a semi-Hilbertian space, i.e., a space induced by a positive operator A, are proved. In particular, among other inequalities, we show that 1/4T#AT + TT#A <= A = 1/4 (2 omega(2)(A) A(T)+ gamma(T)) <= omega(2)(A)(T), where gamma(T) = root(parallel to R-A(T)parallel to(2)(A) - parallel to Z(A)(T)parallel to(2)(A))2 + 4 parallel to R-A(T)Z(A)(T)parallel to(2)(A). Here omega(A)(T) and parallel to T parallel to(A) denote respectively the A-numerical radius and the A-seminorm of an operator T. Also, R-A(T) := T+T#A/2 and I-A(T) := T-T-#A/2i, where T-#A is a distinguished A-adjoint operator of T. Further, some new refinements of the triangle inequality related to parallel to.parallel to(A) are established.