Some numerical radius inequality for several semi-Hilbert space operators

CONDE, CRISTIAN; FEKI, KAIS

LINEAR AND MULTILINEAR ALGEBRA

TAYLOR & FRANCIS LTD

Año: 2023 p. 1 - 18

0308-1087

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space $mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $mathcal{H}$. Here $A$ is not assumed to be invertible. Mainly, if we denote by $omega_A(cdot)$ and $omega(cdot)$ the generalized and the classical numerical radii respectively, we prove that for every $A$-bounded operator $T$ we have$$omega_A(T)=omegaBig(A^{1/2}T(A^{1/2})^{dag}Big),$$where $(A^{1/2})^dag$ is the Moore-Penrose inverse of $A^{1/2}$. In addition, several new inequalities involving $omega_A(cdot)$ 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. 588 (2020) 489-496].