Skip to content
DraconiticResearch
Search Sources API
186,546 studies31 sourcesUpdated 59 min ago
PreprintCC BY 4.0arXiv

Riccati Equations and the Third Problem of Halmos

  • Kui Ji
  • Zhenyue Wang
Open at sourcePDFDOICite

Abstract

Rencently, Fang, Hou, Jiang, and Zhang reduced Halmos's third problem to invertible operators with conjugate bi-geometric form. An operator has such a form if, with respect to an orthogonal decomposition, it admits a representation T = A & C 0 & B , A*, B ∈ B1(Oδ), where Oδ={z∈ℂ:|z|<δ} for some δ>0, and B1(Oδ) denotes the rank-one Cowen--Douglas class. We restrict our attention to the case B=A* with strictly positive coupling C, and settle Halmos's third problem for this class of operators. More precisely,we prove that every such operator T is invertible and that T−1 has a nontrivial closed invariant subspace. This result contributes to the study of Halmos's third problem within the conjugate bi-geometric framework.The proof combines monotone solutions of operator Riccati equations with strong operator convergence to construct invariant subspaces for associated Hamiltonian operator matrices.

Related

  1. Geometric operators and the third problem of Halmos
    Kui Ji, Chunlan Jiang, Zhenyue Wang
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. Dynamic programming principle for controlled systems driven by fractional Brownian motions and related backward stochastic Riccati equations
    Yuecai Han, Yaozhong Hu, Yuhang Li
    arXivPreprintarXiv licenceFull textOptimisation & controlProbabilityarXiv
  3. On the Factored Newton Iteration for Positive-Real Algebraic Riccati Equations via the Matrix Sign Function
    Yiqin Lin, Wenbo Li, Liping Zhou
    MathematicsPeer-reviewedCC BY 4.0General mathematicsCrossref
  4. Stochastic Riccati equations with jumps for indefinite linear-quadratic control: A stochastic value-flow approach
    Kai Ding, Jiaqiang Wen, Jie Xiong +1
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv
  5. Constrained Mean Variance Portfolio Selection under Reflected Factor Dynamics: Stochastic Riccati Equations, Neumann Problems,and a Deep Solver
    Zhecheng Huang, Guojiang Shao, Qi Zhang
    arXivPreprintarXiv licenceFull textOptimisation & controlarXiv

Cite

BibTeXRIS
BibTeX
@misc{ji2026riccati,
  title = {{Riccati Equations and the Third Problem of Halmos}},
  author = {Kui Ji and Zhenyue Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39605},
  eprint = {2610.39605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39605},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Riccati Equations and the Third Problem of Halmos
AU  - Kui Ji
AU  - Zhenyue Wang
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39605
UR  - https://arxiv.org/abs/2610.39605
AB  - Rencently, Fang, Hou, Jiang, and Zhang reduced Halmos's third problem to invertible operators with conjugate bi-geometric form. An operator has such a form if, with respect to an orthogonal decomposition, it admits a representation T = A & C 0 & B , A^*, B ∈ B_1(O_δ), where O_δ={z∈ℂ:|z|<δ} for some δ>0, and B_1(O_δ) denotes the rank-one Cowen--Douglas class. We restrict our attention to the case B=A^* with strictly positive coupling C, and settle Halmos's third problem for this class of operators. More precisely,we prove that every such operator T is invertible and that T^−1 has a nontrivial closed invariant subspace. This result contributes to the study of Halmos's third problem within the conjugate bi-geometric framework.The proof combines monotone solutions of operator Riccati equations with strong operator convergence to construct invariant subspaces for associated Hamiltonian operator matrices.
ER  -