A refinement of A-Buzano inequality and applications to A-numerical radius inequalities

Let A be a positive bounded operator on a Hilbert space H and let ‖T‖A, wA(T), and mA(T) denote the A-operator seminorm, the A-numerical radius, and the A-minimum modulus of an operator T in the semi-Hilbertian space (H,‖⋅‖A), respectively. In this paper, we present new improvements of certain A-Cauchy–Schwarz type inequalities and as applications of our results, we provide refinements of some A-numerical radius inequalities for semi-Hilbertian space operators. It is shown, among other inequalities, thatwA(T)≤(1−12infλ∈C⁡mA2(I−λT))‖T‖A, where I is the identity operator on H. A refinement of the triangle inequality for semi-Hilbertian space operators is also given.