Advanced p-numerical radius bounds through partitioned matrix methodologies

The present investigation develops novel upper limits for the p-numerical radius of linear transformations through sophisticated partitioned matrix methodologies. We derive multiple innovative inequalities governing sums and products of transformations in B ( H ) , with specialized emphasis on positive semidefinite transformations. Our contributions substantially advance beyond previous investigations by Audeh et al., (2024), establishing a unified theoretical framework for p-numerical radius constraints that converge to classical numerical radius results when p → ∞ . We demonstrate that for transformations A , B , X , Y ∈ B ( H ) with X and Y positive semidefinite: w p ( A X 1 / 2 Y 1 / 2 B ∗ ) ≤ 2 2 / p − 2 ‖ Ω ‖ p / 2 + 2 2 / p − 1 w p / 2 ( X 1 / 2 A ∗ B Y 1 / 2 ) where Ω = X 1 / 2 A ∗ A X 1 / 2 + Y 1 / 2 B ∗ B Y 1 / 2 . Our principal results yield superior bounds compared to existing literature and establish the efficacy of partitioned matrix approaches in transformation theory. We analyze particular cases for trace class transformations ( p = 1 ) and Hilbert–Schmidt class ( p = 2 ). Through concrete examples, we substantiate that our generalized inequalities produce sharper constraints than existing results. When p → ∞ , our findings reduce to bounds established in Audeh (2024), confirming that our work constitutes a genuine extension of prior research.