Some numerical radius inequality for several semi-Hilbert space operators

Abstract

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space (Formula presented.), which are bounded with respect to the seminorm induced by a positive operator A on (Formula presented.). Here A is not assumed to be invertible. Mainly, if we denote by (Formula presented.) and (Formula presented.) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have (Formula presented.) where (Formula presented.) is the Moore-Penrose inverse of (Formula presented.). In addition, several new inequalities involving (Formula presented.) 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].