By means of a weaker functional model, we prove the existence of non-trivial closed hyperinvariant subspaces for linear bounded operators generalizing, in particular, a classical theorem of Atzmon and revealing the spectral nature of the hyperinvariant subspaces involved. As an application, we show non-trivial spectral subspaces for Bishop operators on \(L^p[0,1)\), \(1\le p<\infty \), as long as they satisfy Atzmon’s Theorem, providing, in turns, a local spectral decomposition.

中文翻译：

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Atzmon的超不变子空间定理引起的谱分解

通过一个较弱的函数模型，我们证明了线性有界算子的非平凡封闭超不变子空间的存在，特别是对经典的Atzmon定理进行了归纳，并揭示了所涉及的超不变子空间的频谱性质。作为应用程序，我们将在\（L ^ p [0,1）\），\（1 \ le p <\ infty \）上显示Bishop算子的非平凡谱子空间，只要它们满足Atzmon定理，即可提供：依次是局部频谱分解。