Interpolating numerical radius inequalities for matrices (Q6611853)

The main focus of this paper is on proving several new generalized numerical radius inequalities for matrices induced by the unitarily invariant norm, which is defined as\N\[\Nw_{\|\cdot\|}(A) = \max_{\theta\in\mathbb{R}} \|\operatorname{Re}(e^{i\theta}A)\|.\N\]\NFor instance, if \(A,B,X\in M_n(\mathbb{C})\) and \(q\in[0,1]\), the authors show that\N\begin{align*}\N\|AXB^*\|^2 \leq& w_{\|\cdot\|} \left( U^*(qA^*AX + (1-q)XB^*B)V \right) \\\N&\times w_{\|\cdot\|} \left(U^*( (1-q)A^*AX + qXB^*B)V \right),\N\end{align*}\Nfor some unitary matrices \( U, V \) in \( M_n(\mathbb{C}) \). Specifically, if \( X \) is positive semidefinite, the inequality simplifies to:\N\begin{align*}\N\|AXB^*\|^2 \leq &w_{\|\cdot\|} \left( qA^*AX + (1-q)XB^*B \right) \\\N&\times w_{\|\cdot\|} \left( (1-q)A^*AX + qXB^*B \right).\N\end{align*}\NSince \(w_{\|\cdot\|}(A) \leq \|A\|\), this inequality refines the inequality\N\begin{align*}\N\|AXB^*\|^2 \leq &\left\| qA^*AX + (1-q)XB^*B \right\| \\\N&\times \left\| (1-q)A^*AX + qXB^*B \right\|,\N\end{align*}\Nvalid for positive semidefinite matrices, and proven in [\textit{L. Zou} and \textit{Y. Jiang}, J. Math. Inequal. 10, No. 4, 1119--1122 (2016; Zbl 1366.15016)]. This result is only one amongst many that one may find in this interesting paper.