具有非基特征向量族的$C_0$ -群的生成元的预解式和XYZ定理的尖锐性
Resolvent of the generator of the $C_0$-group with non-basis family of eigenvectors and sharpness of the XYZ theorem
摘要：
本文给出了作者最近在《\cite{Sklyar3}》中构造的具有纯虚特征值、聚在$i\infty$上、特征向量的完全极小非基族的$C_0$ -群的生成元类的预解式的一个显式形式。描述了该分解物的生长特性。离散的Hardy不等式是证明相应结果的基石。进一步证明了G.Q.~（10）关于$C_0$ -群的生成元的不变子空间的Riesz基性质的主要结果徐先生~ Yung和H.~ ~ \cite{Xu}，~ \cite{Zwart}中的Zwart是尖锐的。
英文摘要：
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.
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