The invariant subspace problem and Rosenblum operators II
Abstract
Let 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 Kx=∑n=0∞Tnx⊗ en, 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 (H2,S) by the Sobolev disk algebra R(𝔻) and the multiplication operator Mz, whose canonical left inverse B plays the role of S*, and study the two-parameter family Kx,y=∑n=0∞Tnx⊗ B*ny. We prove that T∈ B(R(𝔻)) with r(T)<1 is intransitive whenever there exist a nonzero vector g and a nontrivial Mz-invariant manifold ℳ such that f(T)g has a zero in the closed unit disk for every f∈ℳ; the proof explicitly constructs a nontrivial invariant subspace of T. We further introduce the ideal property and show that every intransitive operator on ℋ is unitarily equivalent to an operator on R(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 ∑n=0∞⟨ Tnξ,η⟩ zn 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.
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BibTeX
@misc{fang2026invariant,
title = {{The invariant subspace problem and Rosenblum operators II}},
author = {Junsheng Fang and Bingzhe Hou and Chunlan Jiang and Yuanhang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11202},
eprint = {2610.11202},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11202},
}
RIS
TY - UNPB
TI - The invariant subspace problem and Rosenblum operators II
AU - Junsheng Fang
AU - Bingzhe Hou
AU - Chunlan Jiang
AU - Yuanhang Zhang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11202
UR - https://arxiv.org/abs/2610.11202
AB - Let 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=∑_n=0^∞T^nx⊗ 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(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=∑_n=0^∞T^nx⊗ B^*ny. We prove that T∈ B(R(𝔻)) with r(T)<1 is intransitive whenever there exist a nonzero vector g and a nontrivial M_z-invariant manifold ℳ such that f(T)g has a zero in the closed unit disk for every f∈ℳ; the proof explicitly constructs a nontrivial invariant subspace of T. We further introduce the ideal property and show that every intransitive operator on ℋ is unitarily equivalent to an operator on R(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 ∑_n=0^∞⟨ T^nξ,η⟩ 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.
ER -