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Partial factorization and reflexivity of operator algebras

  • Fengyang Jia
  • Guoxing Ji
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Abstract

Let ℋ be a separable infinite dimensional Hilbert space and ℬ(ℋ) the algebra of all bounded linear operators on ℋ. A subalgebra A in ℬ(ℋ) has the left (resp.\ right) partial factorization property if for any invertible operator S∈ℬ(ℋ), there exists an isometry (resp.\ a co-isometry) U∈ℬ(ℋ) such that U*S, S−1U∈A. We show that if A is weak operator topology closed with the left (resp.\ right) partial factorization property, then A is the nest algebra associated with its invariant subspace lattice. In particular, if A is transitive, then A=ℬ(ℋ). This gives a positive answer to Question 6.3 raised by B.V.R. Bhat and M. Kumar in \emph{Publ. Res. Inst. Math. Sci.} \textbf{60}(2024), 507--537.

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Cite

BibTeXRIS
BibTeX
@misc{jia2026partial,
  title = {{Partial factorization and reflexivity of operator algebras}},
  author = {Fengyang Jia and Guoxing Ji},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27479},
  eprint = {2609.27479},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27479},
}
RIS
TY  - UNPB
TI  - Partial factorization and reflexivity of operator algebras
AU  - Fengyang Jia
AU  - Guoxing Ji
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27479
UR  - https://arxiv.org/abs/2609.27479
AB  - Let ℋ be a separable infinite dimensional Hilbert space and ℬ(ℋ) the algebra of all bounded linear operators on ℋ. A subalgebra A in ℬ(ℋ) has the left (resp.\ right) partial factorization property if for any invertible operator S∈ℬ(ℋ), there exists an isometry (resp.\ a co-isometry) U∈ℬ(ℋ) such that U^*S, S^−1U∈A. We show that if A is weak operator topology closed with the left (resp.\ right) partial factorization property, then A is the nest algebra associated with its invariant subspace lattice. In particular, if A is transitive, then A=ℬ(ℋ). This gives a positive answer to Question 6.3 raised by B.V.R. Bhat and M. Kumar in \emph{Publ. Res. Inst. Math. Sci.} \textbf{60}(2024), 507--537.
ER  -