$q$-Numerical radius of sectorial matrices and $2\times 2$ 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: $$\frac{\mathrm{\mid }q{\mathrm{\mid }}^2{\cos }^2\alpha }{2}\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }\le w_q^2(A)\le \frac{{(\sqrt{(1-\mathrm{\mid }q{\mathrm{\mid }}^2)(1+2si{n}^2(\alpha ))}+\mathrm{\mid }q\mathrm{\mid })}^2}{2}\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel },$$ 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 $2\times 2$ operator matrices are developed.