$H$ and $k\in\mathbb{N}$. Then:

1.  The space $H_k\left(\{e_n\}\right)=\left\{x=(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n:\: \{c_n\}_{n=1}^{\infty}\in \ell_2(\Delta^k)\right\}$, where $(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n$ denotes a formal series and $\ell_2(\Delta^k)=\left\{\{c_n\}_{n=1}^{\infty}:\:\Delta^k \{c_n\}_{n=1}^{\infty} \in \ell_2\right\},$ is a separable Hilbert space.

2.  The sequence $\{e_n\}_{n=1}^{\infty}$ is dense and minimal in $H_k\left(\{e_n\}\right)$, but it is not uniformly minimal in $H_k\left(\{e_n\}\right)$. Hence $\{e_n\}_{n=1}^{\infty}$ does not form a Schauder basis of $H_k\left(\{e_n\}\right)$.

3.  The operator $A_k:H_k\left(\{e_n\}\right) \supset D(A_k) \mapsto H_k\left(\{e_n\}\right),$ defined by $$A_k x=A_k (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n= (\mathfrak{f})\sum\limits_{n=1}^{\infty} i f(n) \cdot c_{n} e_n,$$ where $\left\{f(n)\right\}_{n=1}^{\infty}\in\mathcal{S}_k$, with domain $$\label{Domain_k}
        D(A_k)=\left\{x= (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n \in H_k\left(\{e_n\}\right):\:  \{f(n) \cdot c_{n}\}_{n=1}^{\infty}\in \ell_2(\Delta^k)\right\},$$ generates the $C_0$-group $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ on $H_k\left(\{e_n\}\right)$, which acts for every $t\in\mathbb{R}$ by the formula $$e^{A_k t} x=e^{A_k t}(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_n e_n =(\mathfrak{f})\sum\limits_{n=1}^{\infty} e^{i t f(n)} c_{n} e_n.$$

Similar results hold for the case of operators with the same spectral behaviour on certain Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$, see [@Sklyar3]. Note that if we take, for example, $f(n)=\sqrt{n},$ $n\in\mathbb{N},$ and define the operator $A_1$ on $H_1\left(\{e_n\}\right)$ as in Theorem \[construction\], then $A_1$ will not generate a $C_0$-semigroup on $H_1\left(\{e_n\}\right)$, see [@Sklyar3], Proposition 7, Proposition 8.

The main objective of the paper is to obtain an explicit form of the resolvent for the class of generators $A_k$ of 