A-numerical radius and A-norm inequalities for semi-Hilbertian space operators

Abstract

ABSTRACT Let (H, 〈 · , · 〉) be a complex Hilbert space and A be a positive bounded linear operator on H. The semi-inner product 〈x, y〉 A : = 〈Ax, y〉, x, y ∈ H, induces a semi-norm on H. Let ω A (T) and denote the A-numerical radius and the A-operator semi-norm of an operator T in semi-Hilbertian space (H, 〈 · , · 〉 A ), respectively. In this paper, some new bounds for the A-numerical radius of operators in semi-Hilbertian space are obtained, which improve the existing ones. In particular, a refinement of the triangle inequality for A-operator semi-norm is also shown.