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

Abstract

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.

arXiv comment34 pages, some results were sharpen, references were added, proofs were complemented