REFINEMENT OF SEMINORM AND NUMERICAL RADIUS INEQUALITIES OF SEMI-HILBERTIAN SPACE OPERATORS

Let H be a complex Hilbert space and A be a non-zero positive bounded linear operator on H. The main aim of this paper is to discuss a general method to develop A-operator seminorm and A-numerical radius inequalities of semi-Hilbertian space operators using the existing corresponding inequalities of bounded linear operators on H. Among many other inequalities we prove that if S, T, X is an element of B-A(H), i.e., if A-adjoint of S,T, X exist, then 2 parallel to S-#A XT parallel to(A) <= parallel to SS#A X + XTT#A parallel to(A). Further, we prove that if T is an element of B-A (H), then 1/4 parallel to(TT)-T-#A + TT#A parallel to(A) <= 1/8(parallel to T + T-#A parallel to(2)(A) + parallel to T - T-#A parallel to(2)(A)) <= 1/8 (parallel to T + T-#A parallel to(2)(A) + parallel to T - T-#A parallel to(2)(A)) + 1/8c(A)(2)(T + T-#A) + 1/8c(A)(2)(T - T-#A) <= omega(2)(A)(T). Here omega(A)(.), c(A)(.) and parallel to.parallel to(A) denote A-numerical radius, A-Crawford number and A-operator seminorm, respectively. (C) Mathematical Institute Slovak Academy of Sciences