$q$-Numerical radius of sectorial matrices and $2 imes 2$ operator matrices

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.

💡 Research Summary

The paper investigates the q‑numerical radius w_q(A) of sectorial matrices, extending and sharpening several known bounds for the classical numerical radius. After recalling basic definitions of numerical range, numerical radius w(T), and q‑numerical radius w_q(T) for operators on a Hilbert space, the authors focus on matrices whose numerical range lies inside a sector S_α = {z∈ℂ : Re z>0, |Im z| ≤ tan α·Re z} with α∈