Pełczyński’s problem concerning the uniqueness of symmetric structure
Let and be separable symmetric Banach sequence spaces, and let and denote the corresponding ideals of compact operators on a separable Hilbert space. Does as Banach spaces imply that with equivalent norms? Equivalently, must an isomorphism between the compact-operator ideals associated with and force the underlying symmetric sequence spaces to coincide up to equivalent norms?
References
Primary source
Additional references
- On Pełczyński's problem concerning the uniqueness of symmetric structure — arXiv — Tianbao Guo, Jinghao Huang
Progress summary
The original classification question remains open; related results concern only narrower isometric or embedding versions.
Pełczyński posed the problem at a Banach-space conference in August 1979: whether an isomorphism between compact-operator ideals associated with symmetric sequence spaces forces the underlying spaces to have equivalent norms. The available evidence records no independently assessed solution.
Known results
- A 2021 preprint proves uniqueness for the narrower isometric analogue of symmetric operator spaces, including the case after a separate argument.
- A 2020 preprint gives non-embedding criteria for symmetric function spaces into compact-operator ideals, extending results of Arazy and Lindenstrauss; it does not claim to solve the uniqueness problem.
Current status (as of October 2026): Pełczyński’s original isomorphic uniqueness problem remains open in the evidence reviewed; related isometric and embedding results are known, but no independently assessed solution is recorded.
Solutions 0
No solutions have been posted yet.