Optimizing Numerical Radius Inequalities via Decomposition Techniques and Parameterized Aluthge Transforms

This manuscript presents substantial refinements to several classical inequalities connecting the numerical radius w(V), spectral radius ρ(V), and operator norm ∥V ∥ for bounded linear operators acting on Hilbert spaces. Building upon inequalities established by Kittaneh [10] and the framework introduced by Yamazaki [11], we develop enhanced bounds through parameterized Aluthge transforms and contemporary decomposition methods. Our key contributions encompass: (1) refined numerical radius bounds that strengthen Kittaneh’s inequality through quantifiable correction terms, (2) parameterized spectral radius inequalities for operator sums and products that significantly improve existing results, and (3) precision-enhanced bounds for commutators and anti-commutators. We provide comprehensive proofs establishing the superiority of our bounds across diverse operator classes. The practical significance of these refinements is demonstrated through applications in numerical linear algebra, where tighter bounds lead to improved convergence estimates for iterative algorithms, and in quantum information theory, where precise operator norm estimates are crucial for error analysis in quantum computing protocols. These refinements yield important theoretical implications in operator theory and matrix analysis, offering substantially tighter estimations of operator spread than previously attainable results.