Let ρ ∈ (0, 2] and let wρ(X) be the ρ-operator radius of a Hilbert space operator X. Using techniques involving the Kronecker product, it is shown that 1 + |1 − ρ| ρ w(X) ≤ wρ(X) ≤ 2 ρ w(X), where w(X) is the numerical radius of X. These bounds for wρ(X) 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 ˜︁ Xt of an operator X, we prove the inequality: wρ(X) ≤ 1 2 wρ(˜︁ Xt) + 1 ρ ∥X∥, for all t ∈ [0, 1]. The derived inequalities extend and generalize several well-known results for the classical operator norm and numerical radius.

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