Rectangular Factorization in the Disk Algebra

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$ and $\overline{\mathbb D}=\{z\in{\mathbb{C}}:|z|\leq1\}$. The disk algebra is \[ A(\mathbb D)=\{f:\overline{\mathbb D}\to{\mathbb{C}}: f\text{ is continuous on }\overline{\mathbb D}\text{ and holomorphic on }\mathbb D\}. \] The supplied problem asks whether every $f\in A(\mathbb D)$ admits a rectangular factorization $f=g*h$ with $g,h\in A(\mathbb D)$. Its intended quantifier structure is \[ \forall f\in A(\mathbb D)\quad\exists g,h\in A(\mathbb D)\quad f=g*h, \] with the additional requirement that this be a rectangular factorization. However, the supplied record defines neither the operation $*$ nor the term "rectangular factorization." Thus this display records the requested formulation but does not specify a fully defined mathematical proposition. The operation and any additional conditions on the factors or equality require clarification from the cited source.

Short English statement

The question asks whether every function continuous on the closed unit disk and holomorphic in its interior has a rectangular factorization into two functions in the same disk algebra. The supplied record leaves both the operation $*$ and the rectangular factorization requirement undefined, so the intended assertion requires clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002504$, title "Rectangular Factorization in the Disk Algebra," in the Oberwolfach Reports Open Problems collection; exact source problem code: catalogue entry. The record gives proposed year 2014 and names no individual proposer.
• Dirichlet Series and Function Theory in Polydiscs (2014), Oberwolfach Reports, report 06/2014, pp. 335--393, DOI: https://doi.org/10.4171/owr/2014/06. This is the report citation supplied as both provenance and the sole literature source checked. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no specific page locating this problem is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the rectangular factorization problem for the disk algebra remained unresolved in the literature checked, with no theorem found proving or refuting factorization for every element. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review concluding that the cleanup was meaning-preserving, self-contained, and added no unsupported source-dependent content. These are supplied verification claims; they do not resolve the missing definitions identified here.