Optimized numerical radius bounds

Abstract

We establish new parameter-dependent upper bounds for the numerical radius of \(2\times 2\) operator matrices. The obtained estimates incorporate mixed product terms such as \(\omega (BC)\) and \(\omega (CB)\), together with norm expressions involving the operator blocks, and refine several known numerical radius inequalities. We further study the associated optimization problems and determine optimal choices of the parameters that yield sharper bounds. As applications, we derive new numerical radius estimates for single operators and recover several known inequalities as special cases.