\sigma_p(A_1)= \{i f(n)\}_{1}^{\infty}$ and $$\rho(A_1)=\left\{ \lambda \in \mathbb{C}: \:\lambda \neq if(n), \: n\in \mathbb{N} \right\}=\mathbb{C} \setminus \sigma(A_1).$$

The proof in the case $k\geq 2$ is based on a combination of ideas of the proof for the case $k=1$ with technical combinatorial elements like in the proof of Theorem 11 from [@Sklyar3] and can be performed similarly to the above.

Define operators $$\widetilde{A_k}:\ell_{p,k}\left(\{e_n\}\right) \supset D\left(\widetilde{A_k}\right) \mapsto \ell_{p,k}\left(\{e_n\}\right)$$ on a class of Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$, see [@Sklyar3] or Section 2, as follows: $$\label{operator on Banach space}
\widetilde{A_k} x=\widetilde{A_k} (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n= (\mathfrak{f})\sum\limits_{n=1}^{\infty} i f(n) \cdot c_{n} e_n,$$ where $\left\{f(n)\right\}_{n=1}^{\infty}\in\mathcal{S}_k$, with domain $$\label{Domain_k p}
    D\left(\widetilde{A_k}\right)=\left\{x= (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n \in \ell_{p,k}\left(\{e_n\}\right):\:  \{f(n) \cdot c_{n}\}_{n=1}^{\infty}\in \ell_p(\Delta^k)\right\}.$$

Then, by virtue of Theorem 16 in [@Sklyar3], $\widetilde{A_k}$ generates the $C_0$-group $\left\{e^{\widetilde{A_k} t} \right\}_{t\in \mathbb{R}}$ on $\ell_{p,k}\left(\{e_n\}\right)$, which acts on $\ell_{p,k}\left(\{e_n\}\right)$ for every $t\in\mathbb{R}$ by the formula $$\label{group}
e^{\widetilde{A_k} t} x=e^{\widetilde{A_k} t}(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_n e_n =(\mathfrak{f})\sum\limits_{n=1}^{\infty} e^{i t f(n)} c_{n} e_n.$$ An explicit form of the resolvent and the description of the spectrum $\sigma\left(\widetilde{A_k}\right)$ of generators $\widetilde{A_k}$ are provided by the following theorem, similar to the Theorem \[Expl\].

\[Expl B\] Let $k\in\mathbb{N}$, $p>1,$ and $\widetilde{A_k}$ be the operator defined above. Then:

1.  $\sigma\left(\widetilde{A_k}\right)= \sigma_p\left(\widetilde{A_k}\right)=\{i f(n)\}_{1}^{\infty}$.

2.  The 