发表机构
Hebei Normal University; Jilin University（河北师范大学; 吉林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究不变子空间问题，通过引入双参数Rosenblum算子族，将Halmos第三问题简化为特定可逆算子情形，证明其逆算子不可传递并构造相关算子。
AI 中文摘要
设$\boldsymbol{\textit{H}}$为可分无限维复Hilbert空间。在本系列第一篇论文中，我们通过Rosenblum算子引入了移位表示算子$K_x=\boldsymbol{\textit{sum}}_{n=0}^{\boldsymbol{\textit{infty}}}T^nx\boldsymbol{\textit{otimes}}e_n$，该算子将谱半径$r(T)<1$的算子$T$与单侧移位算子$S$关联，揭示了其隐藏的解析结构。本文中，我们将$(H^2,S)$替换为Sobolev圆盘代数$R(\boldsymbol{\textit{D}})$及乘法算子$M_z$，其典范左逆$B$扮演$S^*$的角色，研究双参数族$K_{x,y}=\boldsymbol{\textit{sum}}_{n=0}^{\boldsymbol{\textit{infty}}}T^nx\boldsymbol{\textit{otimes}}B^{*n}y$。我们证明：若存在非零向量$g$与非平凡$M_z$不变流形$\boldsymbol{\textit{M}}$，使得对任意$f\boldsymbol{\textit{\u2208}}\boldsymbol{\textit{M}}$，$f(T)g$在闭单位圆盘内有零点，则谱半径$r(T)<1$且属于$B(R(\boldsymbol{\textit{D}}))$的算子$T$是不可传递的；证明过程明确构造了$T$的非平凡不变子空间。我们进一步引入理想性质，证明$\boldsymbol{\textit{H}}$上的每个不可传递算子都酉等价于$R(\boldsymbol{\textit{D}})$上具有该性质的算子，从而将任意算子的不变子空间问题简化为关于具体函数代数的问题。应用包括通过生成函数$\boldsymbol{\textit{sum}}_{n=0}^{\boldsymbol{\textit{\textit{infty}}}}\boldsymbol{\textit{\u27e8}}T^n\boldsymbol{\textit{\u03be}},\boldsymbol{\textit{\u03b7}}\boldsymbol{\textit{\u27e9}}z^n$的有理性刻画不可传递性，以及关于Pearcy问题的新结果。我们的主要应用是Halmos第三问题，该问题已悬而未决五十余年：我们将其简化为具有共轭双几何形式的可逆算子，并证明对每个具有非平凡投影$P$满足$PTP=TP$且$PT(I-P)$秩为1的可逆算子$T$，其逆算子$T^{-1}$是不可传递的。最后，我们给出不可传递的三对角算子，且其存在无限递减的不变子空间链。
英文摘要
Let $\mathcal{H}$ be the separable, infinite-dimensional complex Hilbert space. In the first paper of this series, we introduced, by means of the Rosenblum operators, the shift representation operators $K_x=\sum_{n=0}^{\infty}T^nx\otimes e_n$, which relate an operator $T$ with $r(T)<1$ to the unilateral shift $S$ and reveal its hidden analytic structure. In this paper we replace the pair $(H^2,S)$ by the Sobolev disk algebra $R(\mathbb{D})$ and the multiplication operator $M_z$, whose canonical left inverse $B$ plays the role of $S^*$, and study the two-parameter family $K_{x,y}=\sum_{n=0}^{\infty}T^nx\otimes B^{*n}y$. We prove that $T\in B(R(\mathbb{D}))$ with $r(T)<1$ is intransitive whenever there exist a nonzero vector $g$ and a nontrivial $M_z$-invariant manifold $\mathcal{M}$ such that $f(T)g$ has a zero in the closed unit disk for every $f\in\mathcal{M}$; the proof explicitly constructs a nontrivial invariant subspace of $T$. We further introduce the ideal property and show that every intransitive operator on $\mathcal{H}$ is unitarily equivalent to an operator on $R(\mathbb{D})$ with this property, so that the Invariant Subspace Problem for arbitrary operators reduces to a problem about a concrete function algebra. Applications include a characterization of intransitivity through the rationality of the generating function $\sum_{n=0}^{\infty}\langle T^nξ,η\rangle z^n$ and new results on Pearcy's problem. Our main application is Halmos's third problem, which has remained open for more than fifty years: we reduce it to invertible operators with conjugate bi-geometric form, and we prove that $T^{-1}$ is intransitive for every invertible $T$ admitting a nontrivial projection $P$ with $PTP=TP$ and $PT(I-P)$ of rank one. Finally, we exhibit tridiagonal operators that are intransitive and admit infinite decreasing chains of invariant subspaces.
CommentsThe paper "The Invariant Subspace Problem and Rosenblum Operators I" is available at https://arxiv.org/abs/2506.15270
