Conjecture 4.18 of GLM26

For every separable Hilbert space $\mathcal E$ , every $B\in H^\infty_1(L(\mathcal E))$ , and every $\alpha>-1$ , the associated sub-Bergman spaces satisfy $\mathcal A_\alpha(B)\supset A^2_{\alpha-1,\mathcal H(B(0))}$ and $\mathcal A_\alpha(B^*)\supset A^2_{\alpha-1,\mathcal H(B(0)^*)}$ .

References

Primary source

arXiv

Additional references

Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity — arXiv — Shuaibing Luo, Jiming Shen

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 paper claims to prove both inclusions, while showing that the related rigidity statement fails in infinite dimensions.

Conjecture 4.18 asserts two range inclusions in the sub-Bergman setting. The retrieved material gives no proposer or original date.

September 2026 range-inclusion paper

Shuaibing Luo and Jiming Shen report proofs of both conjectured inclusions for every $\alpha>-1$ . They also construct an infinite-dimensional two-sided inner-function example, showing that the associated rigidity theorem does not extend; the claimed resolution has not been independently verified here.

Current status (as of September 2026): Both range inclusions are claimed proved for every $\alpha>-1$ , while the related rigidity theorem is disproved in the infinite-dimensional setting; the proof and counterexample remain unverified.

Sources

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