Classification of Analytic Hilbert Modules on Strictly Convex Domains

Definitions and mathematical statement

Let $d,m,n$ be positive integers, and let $\Omega\subset{\mathbb{C}}^d$ be a strictly convex domain with smooth boundary. Write \[ L_a^2(\Omega)=\left\{f\in\mathcal O(\Omega):\int_\Omega |f(z)|^2\,{\,\mathrm{d}} V(z)<\infty\right\}, \] where ${\,\mathrm{d}} V$ is Lebesgue volume measure. The source uses the notation $\mathcal A(\Omega)$, calling it the disk algebra. For a bounded domain, the usual interpretation is \[ \mathcal A(\Omega)=\mathcal O(\Omega)\cap C(\overline\Omega), \] with the supremum norm. The source does not explicitly assume that $\Omega$ is bounded or specify a convention for $\mathcal A(\Omega)$ on an unbounded domain; this qualification remains unresolved.

Let $A=(a_{ij})$ be an $n\times m$ matrix with entries in the stated algebra $\mathcal A(\Omega)$. Assume that there exists an integer $r$, with $0\le r\le\min\{m,n\}$, such that \[ \operatorname{rank}A(\zeta)=r\qquad\text{for every }\zeta\in\partial\Omega. \] Thus the rank condition is imposed on the boundary, not throughout $\Omega$. When the entries are bounded multipliers of the Bergman space, define \[ M_A:L_a^2(\Omega)^m\longrightarrow L_a^2(\Omega)^n, \qquad (M_Af)_i=\sum_{j=1}^m a_{ij}f_j. \] In the usual bounded-domain interpretation above, this multiplication map is bounded. The source denotes its cokernel by \[ F=\operatorname{coker}M_A. \] The algebraic quotient $L_a^2(\Omega)^n/\operatorname{ran}M_A$ and the Hilbert-space quotient $L_a^2(\Omega)^n/\overline{\operatorname{ran}M_A}$ agree as quotients only when the range is closed. The supplied record does not state the relevant closed-range conclusion or the complete privilege theorem, so the intended cokernel convention requires clarification.

Requested classification. For such domains and matrices, classify the analytic Hilbert modules $F$ arising in the privilege theorem, and describe their parameter space within the framework of quasi-free Hilbert modules. The supplied original formulation presents these two targets as equivalent.

This is a classification request whose precise equivalence relation and parameter-space structure are not supplied. In particular, the record does not define the privilege condition, specify the intended definition of a quasi-free Hilbert module, explain whether the quotient itself must be quasi-free or is to be described through quasi-free presentations, or choose between unitary module equivalence and bounded invertible module isomorphism. These choices must be resolved before a unique precise classification problem can be stated. No additional interior rank condition, boundedness hypothesis, or equivalence relation is imposed here.

Short English statement

For a smooth strictly convex domain $\Omega\subset{\mathbb{C}}^d$, the source asks for a classification and parameter space of privileged analytic Hilbert modules presented as cokernels of Bergman-space multiplication matrices whose entries lie in $\mathcal A(\Omega)$ and whose boundary rank is constant. The privilege theorem, cokernel convention, equivalence relation, and role of quasi-free modules need clarification. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001225$, source problem code catalogue entry, titled Classification of Analytic Hilbert Modules on Strictly Convex Domains. The record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2009. No individual proposer is supplied.
• Hilbert Modules and Complex Geometry (2009), Oberwolfach Reports, pp. 1047--1100, DOI: 10.4171/owr/2009/20. Supplied source-report link and literature-source link labeled "original report": https://doi.org/10.4171/owr/2009/20.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is supplied source-verification metadata; the report has not been independently consulted for this draft.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and describes the proposed classification beyond stated special cases as a broad open research problem. Those special cases are not specified in the supplied record. This assessment is supplied dated curation, not independently verified; the record explicitly calls for independent verification of time-sensitive status and cited claims.