Generalized A-Numerical Radius of Operators and Related Inequalities

Let A be a non-zero positive bounded linear operator on a complex Hilbert space (H, 〈·, ·〉). Let ωA(T ) denote the A-numerical radius of an operator T acting on the semi-Hilbert space (H, 〈·, ·〉A), where 〈x, y〉A := 〈Ax, y〉 for all x, y ∈ H. Let NA(·) be a seminorm on the algebra of all A-bounded operators acting on H and let T be an operator which admits A-adjoint. Then, we define the generalized A-numerical radius as ωNA(T ) = sup θ∈R NA ( eT + eT ♯A 2 ) , where T ♯A denotes a distinguished A-adjoint of T . We develop several generalized A-numerical radius inequalities from which follows the existing numerical radius and A-numerical radius inequalities. We also obtain bounds for generalized A-numerical radius of sum and product of operators. Finally, we study ωNA(·) in the setting of two particular seminorms NA(·).