article · Journal of Inequalities and Applications
This research develops new generalised Reid-type mathematical inequalities for bounded linear operators on complex Hilbert spaces. By analysing commutation relations and the positivity of two-by-two block operator matrices, the work establishes refined mixed Schwarz-type inequalities alongside Hölder-type bounds for finite sums of operators. These formulations replace standard norm estimates with sharper bounds that incorporate the spectral radius. Furthermore, the findings are applied to deduce new upper bounds for the numerical radius of n-by-n operator matrices. The resulting mathematical bounds generalise and improve upon earlier established results, including recent estimation techniques that incorporate the Moore-Penrose inverse.
Operator theory and matrix bounds are fundamental mathematical tools used across advanced scientific disciplines. By delivering sharper bounds for operator matrices and refining classical inequalities, this theoretical work enhances the mathematical precision available for analysing complex linear systems and operator behaviours.
The abstract does not indicate an application pathway, as it presents purely theoretical mathematical research.
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We establish new generalized Reid-type inequalities for bounded linear operators on complex Hilbert spaces by exploiting the positivity of \(2 \times 2\) block operator matrices and commutation relations. This approach yields refined mixed Schwarz-type inequalities and Hölder-type bounds for finite sums of operators, replacing classical norm estimates with sharper bounds involving the spectral radius. As applications, we derive new upper bounds for the numerical radius of \(n \times n\) operator matrices, which generalize and improve several existing results in the literature, including recent estimates involving the Moore-Penrose inverse.
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DOI: 10.1186/s13660-026-03530-8
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