Perspectives on the ρ-operator radius

Let $\rho \in (0,2]$ and let $w_{\rho }\left(X\right)$ be the ρ-operator radius of a Hilbert space operator X. Using techniques involving the Kronecker product, it is shown that$\frac{1+|1-\rho |}{\rho }w\left(X\right)\le w_{\rho }\left(X\right)\le \frac{2}{\rho }w\left(X\right),$ where $w\left(X\right)$ is the numerical radius of X. These bounds for $w_{\rho }\left(X\right)$ are sharper than those presented by J. A. R. Holbrook. Furthermore, the cases of equality are investigated. We prove that the ρ-operator radius exposes certain operators as projections. We establish new inequalities for the ρ-operator radius, focusing on the sum and product of operators. For the generalized Aluthge transform ${\tilde{X}}_t$ of an operator X, we prove the inequality:$w_{\rho }\left(X\right)\le \frac{1}{2}{w}_{\rho }\left({\tilde{X}}_t\right)+\frac{1}{\rho }\Vert X\Vert ,\text{for all}t\in [0,1].$ The derived inequalities extend and generalize several well-known results for the classical operator norm and numerical radius.