Let $ (\mathcal{K}, \langle \cdot, \cdot \rangle) $ be a complex Hilbert space. Let $ \mathbf{A} $ be a positive (semidefinite) bounded linear operator on $ \mathcal{K} $. We introduce the Euclidean-arithmetic mean $ \mathbf{A} $-numerical radius for a pair of $ \mathbf{A} $-bounded operators $ (\mathbf{U}, \mathbf{V}) $, defined by $ \omega_{\mathbf{A}, e, \varepsilon}(\mathbf{U}, \mathbf{V}) : = \sup\limits_{{\eta \in \Sigma_1^{\mathbf{A}}}} \left((1-\varepsilon) |\langle \mathbf{U}\eta, \eta \rangle_{\mathbf{A}}|^{2} + \varepsilon |\langle \mathbf{V}\eta, \eta \rangle_{\mathbf{A}}|^{2} \right)^{\frac{1}{2}}, $ where $ \varepsilon \in [0, 1] $, $ \langle \eta_1, \eta_2 \rangle_{\mathbf{A}} = \langle \mathbf{A}\eta_1, \eta_2\rangle $ for all $ \eta_1, \eta_2\in \mathcal{K} $, and $ \Sigma_1^{\mathbf{A}} $ denotes the unit $ \mathbf{A} $-sphere. Furthermore, we define the integral Euclidean numerical radius and present several Hermite-Hadamard-type inequalities. Finally, we provide significant applications of our framework by deriving bounds for the Cartesian decomposition of operators and generalizing the Davis-Wielandt radius to a $ p $-arithmetic mean context with $ p \geq 1 $.
Citation: Salma Aljawi, Ahad Hamoud Alotaibi, Silvestru Sever Dragomir, Kais Feki. New inequalities for the $ \mathbf{A} $-joint weighted numerical radius with applications[J]. AIMS Mathematics, 2026, 11(7): 20535-20557. doi: 10.3934/math.2026835
Let $ (\mathcal{K}, \langle \cdot, \cdot \rangle) $ be a complex Hilbert space. Let $ \mathbf{A} $ be a positive (semidefinite) bounded linear operator on $ \mathcal{K} $. We introduce the Euclidean-arithmetic mean $ \mathbf{A} $-numerical radius for a pair of $ \mathbf{A} $-bounded operators $ (\mathbf{U}, \mathbf{V}) $, defined by $ \omega_{\mathbf{A}, e, \varepsilon}(\mathbf{U}, \mathbf{V}) : = \sup\limits_{{\eta \in \Sigma_1^{\mathbf{A}}}} \left((1-\varepsilon) |\langle \mathbf{U}\eta, \eta \rangle_{\mathbf{A}}|^{2} + \varepsilon |\langle \mathbf{V}\eta, \eta \rangle_{\mathbf{A}}|^{2} \right)^{\frac{1}{2}}, $ where $ \varepsilon \in [0, 1] $, $ \langle \eta_1, \eta_2 \rangle_{\mathbf{A}} = \langle \mathbf{A}\eta_1, \eta_2\rangle $ for all $ \eta_1, \eta_2\in \mathcal{K} $, and $ \Sigma_1^{\mathbf{A}} $ denotes the unit $ \mathbf{A} $-sphere. Furthermore, we define the integral Euclidean numerical radius and present several Hermite-Hadamard-type inequalities. Finally, we provide significant applications of our framework by deriving bounds for the Cartesian decomposition of operators and generalizing the Davis-Wielandt radius to a $ p $-arithmetic mean context with $ p \geq 1 $.
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Salma Aljawi, Ahad Hamoud Alotaibi, Silvestru Sever Dragomir, Kais Feki. New inequalities for the $ \mathbf{A} $-joint weighted numerical radius with applications[J]. AIMS Mathematics, 2026, 11(7): 20535-20557. doi: 10.3934/math.2026835
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