Hardy inequality and the construction of infinitesimal operators with non-basis family of eigenvectors

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 complements the results of G.Q. Xu et al. (2005) [28] and H. Zwart (2010) [30] on the Riesz basis property of eigenvectors (eigenspaces) of infinitesimal operators. Our results are extended to the case of operators on some Banach spaces.