Improved Upper Bounds for Numerical Radius Inequalities of Operator Matrices and Their Applications

TL;DR

This paper develops improved upper bounds for the numerical radius of operator matrices using advanced inequalities, with applications in quantum mechanics, differential equations, and fractional calculus.

Contribution

It introduces novel bounds for operator matrices' numerical radii, refining classical inequalities with parameter-dependent techniques and extended Buzano inequalities.

Findings

Enhanced bounds for general operator matrices

Specific inequalities for off-diagonal matrices

Applications to stability analysis in physics

Abstract

This study presents new upper bounds for the numerical radii of operator matrices, with a focus on $n \times n$ and $2 \times 2$ block matrices acting on Hilbert space direct sums. By employing techniques such as the H\"older-McCarthy inequality, Jensen's inequality, Bohr's inequality, and extensions of the Buzano inequality, we derive improved estimates that refine classical results by Kittaneh, El-Haddad, Dragomir, Buzano, and others. Our main contributions include bounds for general operator matrices, specific results for off-diagonal matrices, and inequalities for sums and products of operators. Through detailed comparisons and special cases, we demonstrate that our results enhance existing numerical radius inequalities. A key feature of our work is the use of parameter-dependent refinements with constants derived from extended Buzano-type inequalities. The applications of these results span quantum mechanics, integro-differential equations, and fractional calculus, providing sharper tools for stability analysis and numerical approximation in mathematical physics.