the first part of the Theorem \[res asympt\] may be verified by direct computations and subtle estimates based on the Hardy inequality (\[Hardy\]) similar to those provided by the proof of the Theorem \[Expl\].

For the case of generators of $C_0$-groups acting on a class of Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$, we have the following result, analogous to the first part of the Theorem \[res asympt\].

\[res asympt p\] Let $k\in\mathbb{N}$, $p>1,$ and $\widetilde{A_k}:\ell_{p,k}\left(\{e_n\}\right) \supset D\left(\widetilde{A_k}\right) \mapsto \ell_{p,k}\left(\{e_n\}\right)$ be the operator defined by (\[operator on Banach space\]), (\[Domain\_k p\]), see also Theorem \[Expl B\].

Then for every $a>0$ there exists a constant $C>0$ such that $$\label{r est p}
\left\|\left(\widetilde{A_k}- \lambda I \right)^{-1}\right\|\leq \frac{C}{\left|\Re(\lambda)\right|^{k+1}}\quad\text{for all}\quad \lambda: 0<\left|\Re(\lambda)\right|<a,$$ $$\label{r est 2 p}
\left\|\left(\widetilde{A_k}- \lambda I \right)^{-1}\right\|\leq C\quad\text{for all}\quad \lambda:\left|\Re(\lambda)\right| \geq a.\qquad\qquad\:$$

Denote by $\left\{e^{\widetilde{A_k} t} \right\}_{t\in \mathbb{R}}$ the $C_0$-group corresponding to the operator $\widetilde{A_k}$, see Theorem 16 of [@Sklyar3]. Then, by virtue of Proposition 17 from [@Sklyar3], there exists a polynomial $\mathfrak{p}_k$ with positive coefficients such that $$\deg \mathfrak{p}_k =k$$ and for every $t\in\mathbb{R}$ we have $$\left\|e^{\widetilde{A_k} t}\right\| \leq \mathfrak{p}_k (|t|).$$ Therefore the required estimates follow from the formula for representation of the resolvent (\[representation res\]), see also Theorem 2.1 of [@Eisner1].

