  Some special Hilbert spaces are introduced to present the class of
infinitesimal operators with complete minimal non-basis family of eigenvectors.
The discrete Hardy inequality plays an important role in the proposed approach.
The construction complement the results of G.Q.~Xu et al.~\cite{Xu} (2005) and
H.~Zwart~\cite{Zwart} (2010) on the Riesz basis property of eigenvectors
(eigenspaces) of infinitesimal operators. Our results are extended to the case
of operators on some Banach spaces.
