通过双盘哈代空间上的通用托普利兹算子论不变子空间问题

João Marcos R. do Carmo, Marcos S. Ferreira
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摘要

希尔伯特空间的不变子空间问题(ISP)询问是否每个有界线性算子都有一个非三维封闭不变子空间。由于普遍算子的存在(在罗塔的意义上),ISP 可以通过证明普遍算子的每个最小不变子空间都是一维来解决。在这项工作中(T^{*}_{\varphi}|_{M}\)有一个非三维子空间的条件。其中 \(M\subset H^{2}({\mathbb {D}}^{2})\ 是托普利兹算子 \(T_{\varphi }^{*}\) 的不变子空间)是符号 \(\varphi \in H^{infty }({\mathbb {D}}^{2})\) 所诱导的双盘 \(H^{2}({\mathbb {D}}^{2})\) 上的哈代空间的不变子空间。然后,我们利用这一事实得到 ISP 为真的充分条件。
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On the Invariant Subspace Problem via Universal Toeplitz Operators on the Hardy Space Over the Bidisk

The invariant subspace problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this work, we obtain conditions for \(T^{*}_{\varphi }|_{M}\) to have a non-trivial subspace where \(M\subset H^{2}({\mathbb {D}}^{2})\) is an invariant subspace of the Toeplitz operator \(T_{\varphi }^{*}\) on the Hardy space over the bidisk \(H^{2}({\mathbb {D}}^{2})\) induced by the symbol \(\varphi \in H^{\infty }({\mathbb {D}})\). We then use this fact to obtain sufficient conditions for the ISP to be true.

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