A generalization of the concept of sectorial operator

Let $B$ be a Banach space and $G\colon[0,+\infty)\to(0,+\infty)$ a non-increasing function such that $G(t)\to0$ as $t\to\infty$ and $1/G$ is a Lipschitz function on $[0,+\infty)$.
A linear operator $T\colon D(T)\subset B\to B$ is said to be $G$-sectorial if there exist constants $a\in\mathbb R$ and $\varphi\in(0,\pi/2)$ such that the spectrum of $T$ lies in the set
$$ S_{a,\varphi}:=\{z\in\mathbb C\mid z\ne a,\ \lvert\arg(z-a)\rvert<\varphi\} $$
and
$$ \text{there exists } M>0\quad \text{such that } \|R_\lambda(T)\|\le MG(|\lambda-a|)\text{ for }\lambda\notin S_{a,\varphi}, $$
where $R_\lambda(T)$ is the resolvent of the operator $T$.
The properties of the operator exponential and fractional powers of a $G$-sectorial operator are analysed alongside the question of the unique solubility of the Cauchy problem for the linear differential operator with $G$-sectorial operator-valued coefficient.
Bibliography: 8 titles.