Let $Au_i=u_{i+1}-(2-\beta)u_i+u_{i-1}$ whith $u\in \ell^2=\{(u_i)_{i\in \mathbb Z}, u_i\in \mathbb R:\sum_{i\in \mathbb Z}u^2_i<+\infty\}; \beta>0$. How to compute $||A^{-1}||$ or estimate it? Is it a sectorial operator of type $(\omega, \theta)$ with $\omega <0$?.
$A$ is a sectorial of type $(\omega, \theta)$, that is, there exist $\omega\in\mathbb R, \theta\in (0,\frac {\pi} 2), M>0$ such that its resolvent lies in $\mathbb C\backslash \Sigma_{\omega, \theta}$ and $$ \|(\lambda I - A)^{-1}\|\le \frac M {|\lambda - \omega|}, \;\lambda \not\in \Sigma_{\omega,\theta}, $$ here $$ \Sigma_{\omega,\theta}=\{\omega+\lambda: \lambda\in\mathbb C, |\text{arg}(-\lambda)|<\theta\}. $$