Matrix Young numerical radius inequalities (Q2853311)

Let \(T\) be a bounded linear operator on a Hilbert space \(\mathcal{H}\). Then the numerical radius of \(T\) is defined by NEWLINE\[NEWLINE\omega (T) = \sup \{|\langle Tf, f \rangle | : f \in \mathcal{H}, \|f\| = 1\}.NEWLINE\]NEWLINE The main result of the paper states that for \(p > q > 1\) such that \(\frac{1}{p} + \frac{1}{q} = 1\), and for \(A \in M_n(\mathbb{C})\), a non scalar strictly positive matrix such that \(1 \in \sigma(A)\), there exists \(X \in M_n(\mathbb{C})\) such that NEWLINE\[NEWLINE\omega(AXA) > \omega (\frac{1}{p} A^p X + \frac{1}{q} X A^q).NEWLINE\]NEWLINEThe proof uses the Schur product and the Schur complement techniques of matrices. The authors also explore some general inequalities concerning the numerical radius for matrices.