($${\varepsilon}$$, $$\boldsymbol{\mathcal{A}}$$)-Numerical Radius of Operators and Related Inequalities
Najla A. Altwaijry, Sever S Dragomir, Kais Feki, Hong Qiao · Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) · 2025
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.