Abstract
The concept of the weighted \(\mathcal{A}\) -numerical radius was recently defined, where \(\mathcal{A}\) is assumed to be a positive operator. In this paper, we introduce another weighted \(\mathcal{A}\) -numerical radius, denoted by \(\omega_{(\varepsilon,\mathcal{A})}\left(\cdot\right)\) , for operators in semi-Hilbert spaces. We establish some basic properties and inequalities for \(\omega_{(\varepsilon,\mathcal{A})}\left(\cdot\right)\) , which generalize earlier results about \(\omega_{\mathcal{A}}(\cdot)\) . Specifically, we derive new identities for the \(\mathcal{A}\) -numerical radius and provide further comparisons between the \(\mathcal{A}\) -numerical radius and the operator \(\mathcal{A}\) -seminorm of weighted real and weighted imaginary parts. Additionally, we utilize Boas–Bellman type inequalities in the context of semi-Hilbert spaces to derive upper bounds for \(\omega_{(\varepsilon,\mathcal{A})}\left(\cdot\right)\) . Several applications are also discussed.
