$q$-Numerical radius of sectorial matrices and $2 \times 2$ operator matrices
Jyoti Rani, Arnab Patra
TL;DR
The paper develops refined $q$-numerical radius bounds for sectorial matrices, establishing the sharp two-sided inequality $\frac{|q|^2\cos^2\alpha}{2} \|A^*A+AA^*\| \le w_q^2(A) \le \frac{(\sqrt{(1-|q|^2)(1+2\sin^2(\alpha))}+|q|)^2}{2} \|A^*A+AA^*\|$ for $A\in \prod_{s,\alpha}^n$, and connects these bounds to $w(A)$ and $\|A\|$ via corollaries. It also extends the analysis to non-integral powers, providing relations between $w_q(A^t)$ and $w_q^t(A)$ for $t\in[0,1]$, and to $A^{-t}$, thereby generalizing known bounds from the classical numerical radius. Furthermore, the work develops estimations for the $q$-numerical radius of off-diagonal $2\times2$ operator matrices and derives both upper and lower bounds for matrices of the form $\begin{pmatrix}0 & X\\ Y & 0\end{pmatrix}$, highlighting invariances under unitary similarities and shedding light on the roles of $X,Y$ (including when they are positive or sectorial). Overall, the paper provides tighter, parameter-dependent bounds that refine existing results and broaden applicability to operator matrices.
Abstract
This article focuses on several significant bounds of $q$-numerical radius $w_q(A)$ for sectorial matrix $A$ which refine and generalize previously established bounds. One of the significant bounds we have derived is as follows: \[\frac{|q|^2\cos^2α}{2} \|A^*A+AA^*\| \le w_q^2(A)\le \frac{\left(\sqrt{(1-|q|^2)\left(1+2sin^2(α)\right)}+ |q|\right)^2}{2} \|A^*A+AA^*\|,\] where $ A $ is a sectorial matrix. Also, upper bounds for commutator and anti-commutator matrices and relations between $w_q(A^t)$ and $w_q^t(A)$ for non-integral power $t\in [0,1]$ are also obtained. Moreover, a few significant estimations of $q$-numerical radius of off-diagonal $2\times2$ operator matrices are developed.