To describe the growth properties of the resolvent of operators from Theorem \[construction\] (see also Theorem \[Expl\]) and Theorem \[Expl B\] we use Proposition 12 from [@Sklyar3] on the polynomial boundedness of the constructed $C_0$-groups $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ on $H_k\left(\{e_n\}\right)$ and apply results from [@Eisner1], [@Eisner2], [@Malejki].

\[res asympt\] Let $k\in\mathbb{N}$ and $A_k$ be the operator from Theorem \[construction\].

Then the following assertions are true:

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

2.  There exists a constant $M>0$ such that for every $a>0$ and all $x,y\in H_k\left(\{e_n\}\right)$ we have $$\int\limits_{-\infty}^{\infty} \left|\bigl<\left(A_k\pm (a+is) I \right)^{-2}x,y\bigr> \right| ds\leq \frac{M}{a} \left(1+\frac{1}{a^{2k}} \right)\|x\|\|y\|.$$

3.  There exists a constant $K>0$ such that for every $a>0$ and all $x,y\in H_k\left(\{e_n\}\right)$ we have $$\int\limits_{-\infty}^{\infty} \left\| \left(A_k\pm (a+is) I \right)^{-1}x\right\|^2 ds \leq \frac{K}{a} \left(1+\frac{1}{a^{2k}} \right)\|x\|^2,$$ and $$\int\limits_{-\infty}^{\infty} \left\| \left(A_k^{\ast}\pm (a+is) I \right)^{-1}y\right\|^2 ds \leq \frac{K}{a} \left(1+\frac{1}{a^{2k}} \right)\|y\|^2.$$

1\. Let $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ be the $C_0$-group corresponding to the operator $A_k$, see Theorem \[construction\]. Then, by virtue of Proposition 12 from [@Sklyar3], $C_0$-groups $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ grow 