where $(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n$ also denotes a formal series and $$\ell_p(\Delta^k)=\left\{\{c_n\}_{n=1}^{\infty}:\:\Delta^k \{c_n\}_{n=1}^{\infty} \in \ell_p\right\},$$ is a separable Banach space, isomorphic to $\ell_p$, see [@Sklyar3]. If $p>1$, then the sequence $\{e_n\}_{n=1}^{\infty}$ is dense and minimal in $\ell_{p,k}\left(\{e_n\}\right)$, $p>1,$ $k\in\mathbb{N}$, but it is not uniformly minimal in $\ell_{p,k}\left(\{e_n\}\right)$, so $\{e_n\}_{n=1}^{\infty}$ does not form a Schauder basis of $\ell_{p,k}\left(\{e_n\}\right)$. Using similar arguments we obtain the following result, analogous to Theorem \[sharp\].

\[sharp 2\] Let $k\in\mathbb{N}$ and $\{e_n\}_{n=1}^{\infty}\subset \ell_{p,k}\left(\{e_n\}\right)$, $p\geq 1,$ be a sequence defined above. Suppose that $\{A_n\}_{n=1}^{\infty}$ is an arbitrary decomposition of $\mathbb{N}$ into disjoint sets. Then $\left\{\overline{Lin} \{e_j\}_{j\in A_n}\right\}_{n=1}^{\infty}$ does not form a Schauder decomposition of $\ell_{p,k}\left(\{e_n\}\right)$.

An explicit form of the resolvent of the class of generators of $C_0$-groups
============================================================================

Recall that if $p>1$ and $a_n\geq0$ for $n\in\mathbb{N}$, then the discrete Hardy inequality states that $$\label{Hardy}
    \sum\limits_{n=1}^{\infty} \left(\frac{1}{n} \sum\limits_{k=1}^{n} a_k\right)^p < \left(\frac{p}{p-1}\right)^p \sum\limits_{n=1}^{\infty} a_n^p$$ with the exception of the case when $a_n=0$ for all $n\in\mathbb{N}$. Moreover, the constant $\left(\frac{p}{p-1}\right)^p$ is the best possible.

The following theorem is a central result of the present paper. It provides an explicit form of the resolvent for the class of generators $A_k:H_k\left(\{e_n\}\right) \supset D(A_k) \mapsto H_k\left(\{e_n\}\right)$, $k\in\mathbb{N}$, of $C_0$-groups from Theorem \[construction\] and the description 