Some numerical radius inequality for several semi-Hilbert space operators

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space H, which are bounded with respect to the seminorm induced by a positive operator A on H. Here A is not assumed to be invertible. Mainly, if we denote by omega(A)(.) and omega(.) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have omega(A)(T) = omega(A(1/2)T(A(1/2))(dagger)), where (A(1/2))(dagger) is the Moore-Penrose inverse of A(1/2). In addition, several new inequalities involving omega(A)(.) for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489-496].