Skip to content
DraconiticResearch
Search Sources API
181,686 studies31 sourcesUpdated 18 min ago
PreprintarXiv licencearXiv

On Pełczyński's problem concerning the uniqueness of symmetric structure

  • Tianbao Guo
  • Jinghao Huang
Open at sourcePDFDOICite

Abstract

Let E and F be separable symmetric Banach sequence spaces, and let CE and CF be the corresponding ideals of compact operators on a separable Hilbert space. We prove that if CF is isomorphic to a complemented subspace of SE, then F=ℓ2. As a consequence, if CE is isomorphic to CF, then E=F with equivalent norms. This answers Pełczyński's 1979 question concerning the uniqueness of symmetric structure. We also strengthen Arazy's result by showing that if E has finite cotype and CF embeds isomorphically into SE, then F=ℓ2. For reflexive F≠ℓ2 with non-trivial Boyd indices, the embedding theorems of Lindenstrauss and Szankowski yield a separable reflexive symmetric sequence space E with trivial cotype such that CF embeds isomorphically into SE. This gives negative answers to several questions raised by Arazy in 1981.

Related

  1. On a question by Firey concerning uniqueness
    Carlos Cabezas-Moreno
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  2. On a problem concerning ℤ3-Ramsey numbers
    Cheng Chi, Jialin He
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  3. On a problem of Ma and Chen concerning minimal additive complements
    Shi-Qiang Chen, Quan-Hui Yang
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. On Alon's problem concerning the difference between connected domination number and domination number
    Chengli Li, Bo Zhou
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. On a problem of A. N. Skiba and L. A. Shemetkov concerning Sn-closed subformations of a one-generated formation
    Chenxiao Lin
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{guo2026pełczyńskis,
  title = {{On Pełczyński's problem concerning the uniqueness of symmetric structure}},
  author = {Tianbao Guo and Jinghao Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09923},
  eprint = {2610.09923},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09923},
}
RIS
TY  - UNPB
TI  - On Pełczyński's problem concerning the uniqueness of symmetric structure
AU  - Tianbao Guo
AU  - Jinghao Huang
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09923
UR  - https://arxiv.org/abs/2610.09923
AB  - Let E and F be separable symmetric Banach sequence spaces, and let C_E and C_F be the corresponding ideals of compact operators on a separable Hilbert space. We prove that if C_F is isomorphic to a complemented subspace of S_E, then F=ℓ_2. As a consequence, if C_E is isomorphic to C_F, then E=F with equivalent norms. This answers Pełczyński's 1979 question concerning the uniqueness of symmetric structure. We also strengthen Arazy's result by showing that if E has finite cotype and C_F embeds isomorphically into S_E, then F=ℓ_2. For reflexive F≠ℓ_2 with non-trivial Boyd indices, the embedding theorems of Lindenstrauss and Szankowski yield a separable reflexive symmetric sequence space E with trivial cotype such that C_F embeds isomorphically into S_E. This gives negative answers to several questions raised by Arazy in 1981.
ER  -