Intermediate Bergman Backward-Shift Invariant Subspaces

Definitions and mathematical statement

Extracted interpolation question. For the backward shift $B$ on a Bergman space $A^p$, if $M_1\subsetneq M_2$, does there exist a $B$-invariant subspace $M$ with $M_1\subsetneq M\subsetneq M_2$? In the usual invariant-subspace interpretation, all three subspaces are closed linear subspaces of $A^p$, and $B(M_j)\subseteq M_j$. The source does not state these endpoint hypotheses or any restriction on $M_2/M_1$.

A conventional model is $A^p(D)=\{f\text{ holomorphic on }D:\int_D|f|^p\,dA<\infty\}$, where $D$ is the unit disc and $dA$ area measure. The source does not specify the range of $p$ or weights. One common backward shift is $Bf(z)=(f(z)-f(0))/z$, with the removable value $Bf(0)=f'(0)$. On the Hilbert Bergman space, the adjoint of multiplication by $z$ is a different weighted backward shift: for $f(z)=\sum_{n\ge0}a_nz^n$, it is $\sum_{n\ge0}\frac{n+1}{n+2}a_{n+1}z^n$. These operators must not be identified without the source's convention.

Missing essential qualification. A literal assertion for every strict nested pair cannot hold if $\dim(M_2/M_1)=1$, because then no intermediate linear subspace exists. In either of the conventional shifts above, $M_1=\operatorname{span}\{1\}$ and $M_2=\operatorname{span}\{1,z\}$ are closed invariant subspaces with one-dimensional quotient. Thus an omitted restriction excluding such pairs is essential to the intended open interpolation problem. A condition such as an infinite-dimensional quotient would define a different, qualified question, but it is not inserted as if supplied. Once the endpoint and closedness conventions are fixed, the intermediate-space problem corresponds to asking for a nonzero proper closed invariant subspace of the induced operator on $M_2/M_1$; this restates the need for the quotient hypotheses rather than settling that qualified problem.

The supplied triage checked 2026-08-21 labels the Bergman backward-shift interpolation problem open, reporting structural partial results and no located general intermediate-subspace theorem. That cannot certify the literal unrestricted wording in the face of the one-dimensional obstruction. The original source's missing pair restrictions and operator convention must be recovered before applying this dated status assessment.

Short English statement

The intended problem asks for a closed invariant space strictly between two given invariant spaces. Essential restrictions are missing: a one-dimensional quotient cannot have an intermediate space. The meaning of backward shift and the range of $p$ also need source verification.

Sources

• Local UnsolvedMath registry, permanent ID 30000291, code catalogue entry; Intermediate Bergman Backward-Shift Invariant Subspaces, Oberwolfach Reports Open Problems.
• Mini-Workshop: Operators on Spaces of Analytic Functions (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/36, pp. 2043--2090. https://doi.org/10.4171/owr/2005/36. Proposed year and year tag: 2005; no proposer named.
• Supplied literature label original OWR problem: https://doi.org/10.4171/owr/2005/36.
• Supplied literature label later OWR open-problem list: https://ems.press/content/serial-article-files/46010.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports checking the official report, not independently checked here.