Resolvent of the generator of the $C_0$-group with non-basis family of eigenvectors and sharpness of the XYZ theorem

TL;DR

This paper derives an explicit resolvent form for certain $C_0$-group generators with non-basis eigenvectors, analyzes their growth, and demonstrates the sharpness of a classical Riesz basis theorem.

Contribution

It provides a new explicit resolvent formula for generators with non-basis eigenvectors and confirms the optimality of a key Riesz basis result.

Findings

Explicit resolvent form for generators with non-basis eigenvectors

Growth properties of the resolvent described

Sharpness of the Riesz basis theorem established

Abstract

The paper presents an explicit form of the resolvent for the class of generators of $C_0$-groups with purely imaginary eigenvalues, clustering at $i\infty$, and complete minimal non-basis family of eigenvectors, constructed recently by the authors in~\cite{Sklyar3}. The growth properties of the resolvent are described. The discrete Hardy inequality serves as the cornerstone for the proofs of the corresponding results. Moreover, it is shown that the main result on the Riesz basis property for invariant subspaces of the generator of the $C_0$-group, obtained a decade ago by G.Q.~Xu, S.P.~Yung and H.~Zwart in~\cite{Xu},~\cite{Zwart}, is sharp.