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

TL;DR

This paper introduces special Hilbert spaces to construct infinitesimal operators with a complete minimal non-basis family of eigenvectors, utilizing Hardy inequalities, and extends these results to Banach spaces.

Contribution

It presents a novel method for constructing infinitesimal operators with non-basis eigenvector families using Hardy inequalities, extending prior work to Banach spaces.

Findings

Constructed operators with non-basis eigenvector families

Extended Hardy inequality applications to Banach spaces

Provided new insights into eigenvector basis properties

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.