Shimorin’s Beurling-type conjecture

Let $A^2_\alpha$ be the weighted Bergman space on $\mathbb{D}$ , with $\alpha>-1$ , and let $M\subseteq A^2_\alpha$ be invariant under the multiplication operator $S f(z)=z f(z)$ . Define the wandering subspace $W(M)=M\ominus S M$ . The conjecture asks whether every such invariant subspace satisfies $M=\overline{\operatorname{span}}\{S^n w:n\ge 0,\ w\in W(M)\}$ . The claimed sharp statement is that this holds for every invariant $M$ if and only if $-1<\alpha\le 1$ ; for every $\alpha>1$ , there exists a finite set $A\subset\mathbb{D}$ such that the zero-based invariant subspace $I_A$ fails this equality.

References

Primary source

arXiv

Additional references

The Critical Value for the Beurling-Type Theorem in Weighted Bergman Spaces — arXiv — Zhaopeng Lin, Shibo Xu, Tao Yu

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 unrefereed preprint claims to settle the conjecture by identifying exactly when the proposed property fails, but the result has not been independently checked.

Shimorin’s conjecture concerns the wandering-subspace property for invariant subspaces of weighted Bergman shifts, with parameter $\alpha$ . The new claim says the property holds at and below the threshold $\alpha=1$ but fails for every $\alpha>1$ .

Known results

Shimorin proved the property for $-1<\alpha<0$ and $0<\alpha\le 1$ .
Aleman, Richter, and Sundberg proved it for the unweighted Bergman shift.
An earlier result established failure for $\alpha\ge 1.04$ .
Liu’s 2022 work concerns reducing subspaces for multiplicity- $N$ shifts, not the full conjecture.

September 2026 preprint

Zhaopeng Lin, Shibo Xu, and Tao Yu claim to construct, for every $\alpha>1$ , a finite zero set whose invariant subspace fails the wandering-subspace property, establishing the sharp threshold $\alpha=1$ . The preprint is unrefereed, and the broader scan found no independent verification or correction.

Current status (as of September 2026): The conjecture is claimed settled by an unrefereed preprint, but its threshold result remains unverified; the cases $\alpha\le 1$ are established by earlier work.

Sources

No solutions have been posted yet.