The paper presents an explicit form of the resolvent and characterisation of the spectrum 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 [28]. The discrete Hardy inequality serves as the cornerstone for the proofs of the corresponding results. Furthermore, it is shown that the main result on the Riesz basis property for invariant subspaces of the generator of the $C_0$-group (the XYZ theorem), obtained a decade ago by G. Q. Xu, S. P. Yung and H. Zwart in [31] and [32], is sharp.

中文翻译：

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具有非基本特征向量族和XYZ定理的清晰度的$ C_0 $组生成器的解析

本文针对具有纯虚构特征值的$ C_0 $ -groups生成器类别，给出了一种清晰的分解形式和频谱特征，其聚类为$ i \ infty $，并构造了完整的最小非基本特征向量族由[28]中的作者撰写。离散的Hardy不等式是证明相应结果的基础。此外，它还表明$ G_0 $ -group生成器的不变子空间的Riesz基性质的主要结果（XYZ定理）是十年前由GQ Xu，SP Yung和H.Zwart在[31]中获得的。 ]和[32]是尖锐的。