c_n \phi_n.$$ Then we define the operator $A:H\supset D(A)\mapsto H$ as follows, $$Ax=A \sum\limits_{n=1}^{\infty} c_n \phi_n = -\sum\limits_{n=1}^{\infty} n c_n \phi_n,$$ where $$D(A)=\left\{x=\sum\limits_{n=1}^{\infty} c_n \phi_n \in H:\: \sum\limits_{n=1}^{\infty} n c_n \phi_n \in H\right\}.$$

It can be easily shown that $A$ generates the $C_0$-semigroup on $H$, the spectrum of $A$ is pure point and consists of simple eigenvalues $-n$, $n\in\mathbb{N}$, with corresponding eigenvectors $\{\phi_n\}_{n=1}^{\infty}$, see, e.g., [@Haase]. Finally, it is not hard to prove that, if $\{A_n\}_{n=1}^{\infty}$ is a decomposition of $\mathbb{N}$ into disjoint sets with at most $K$ elements in each of them, such that (\[disj sets\]) holds, then $$\left\{\overline{Lin} \{\phi_j\}_{j\in A_n}\right\}_{n=1}^{\infty}$$ does not form a Riesz basis of subspaces of $H$.

For our construction of generators of $C_0$-groups with complete minimal non-basis family of eigenvectors in special classes of Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$ (see Theorem 16 in [@Sklyar3]), we have a result similar to the Theorem \[sharp\]. Here $\{e_n\}_{n=1}^{\infty}$ denotes an arbitrary symmetric basis of an initial Banach space $\ell_p$, $p\geq 1.$ Recall that Schauder basis $\{e_n\}_{n=1}^{\infty}$ is called symmetric provided any its permutation $\{e_{\theta(n)}\}_{n=1}^{\infty}$, $\theta(n): \mathbb{N}\mapsto \mathbb{N}$, also forms a Schauder basis, equivalent to $\{e_n\}_{n=1}^{\infty}$. For any $p\geq 1$ and $k\in\mathbb{N}$ the space $$\ell_{p,k}\left(\{e_n\}\right)=\left\{x=(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n:\: \{c_n\}_{n=1}^{\infty}\in \ell_p(\Delta^k)\right\},$$ 