Bloch Spectra and Hilbert-Space Characterizations

Definitions and mathematical statement

The source asks whether every complex Banach space $X$ satisfying \[ \Lambda_{\mathrm{Bloch},\ell^1}(X)=\ell^{(2,1)} \] must be isomorphic to a Hilbert space. The source does not define the invariant on the left, the sequence-space convention on the right, or the meaning of equality. In particular, the notation $\ell^{(2,1)}$ must not be assigned a Lorentz or mixed/block norm merely from its typography. The title "Bloch Spectra" does not supply the missing construction.

The conclusion has an unambiguous standard interpretation: there should exist a complex Hilbert space $H$ and a bounded complex-linear bijection $T:X\to H$ with bounded inverse. Equivalently, $X$ should admit a positive Hermitian inner product $[\cdot,\cdot]$ and constants $0<c\leq C<\infty$ such that \[ c\|x\|_X^2\leq[x,x]\leq C\|x\|_X^2\qquad(x\in X). \] This asks for an equivalent Hilbertian norm, not that the given norm already satisfy an inner-product identity.

Thus the intended universal implication is recorded with its exact displayed hypothesis, while its mathematical verification requires the original definitions, index/weight conventions, and whether equality means equality of sequence sets, normed identification, or norm equivalence. No substitute for $\Lambda_{\mathrm{Bloch},\ell^1}(X)$ is invented.

The supplied literature triage dated 21 August 2026 discusses sufficiency for $L^2$, counterexamples among symmetric spaces with trivial Boyd indices, and an incomplete classification of rearrangement-invariant spaces. It does not explain how that classification relates to the displayed Bloch-invariant implication for arbitrary complex Banach spaces. The reported partial results are therefore retained as a possible curation-context mismatch, not asserted as theorems about this undefined hypothesis. The supplied status is partially solved, with the exact scope unresolved.

Short English statement

The target is an equivalent Hilbert norm for a complex Banach space satisfying the displayed Bloch sequence-space identity. Both sides of that hypothesis lack definitions in the record, and the supplied literature assessment discusses a different apparent classification framework. The gap is explicit rather than filled by guessing the notation.

Sources

• Local UnsolvedMath registry, permanent ID $30000102$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; verification only removes a question label and balances the TeX, without supplying invariant definitions.
• "Spectral Theory in Banach Spaces and Harmonic Analysis (2005)," Oberwolfach Reports, pp. 1883--1970, DOI 10.4171/owr/2004/36: https://doi.org/10.4171/owr/2004/36. Proposed year and DOI refer to 2004, while the citation label says 2005.
• Supplied "de Pagter--Sukochev": https://doi.org/10.1016/S0022123603003860. The other supplied source is the original report DOI above. No full invariant definition or explicit connecting theorem is included.
• Literature checked 21 August 2026; supplied status partially_solved, with an assessment about rearrangement-invariant spaces and Boyd indices. Registry created 25 August 2026 and updated 16 September 2026. This dated triage requires independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.