Unconditional Reproducing-Kernel Bases in Model Spaces

Definitions and mathematical statement

The supplied question concerns whether every model space admits an unconditional basis of reproducing kernels. The Hardy-space domain and the permitted locations of the kernels are not specified in the record. The following is a precise formulation under the explicit convention of the unit disc and interior evaluation points; this convention requires source confirmation.

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$, and define \[ H^2(\mathbb D)=\left\{f(z)=\sum_{n=0}^{\infty}a_nz^n:\sum_{n=0}^{\infty}|a_n|^2<\infty\right\}, \qquad \|f\|_{H^2}^2=\sum_{n=0}^{\infty}|a_n|^2. \] An inner function $\Theta$ is a bounded analytic function on $\mathbb D$ whose radial boundary values have modulus one almost everywhere on the unit circle with respect to arc length. Its model space is \[ K_\Theta=H^2(\mathbb D)\ominus\Theta H^2(\mathbb D). \] For $\lambda\in\mathbb D$, its reproducing kernel is \[ k_\lambda^\Theta(z)=\frac{1-\overline{\Theta(\lambda)}\Theta(z)}{1-\overline\lambda z}, \qquad f(\lambda)=\langle f,k_\lambda^\Theta\rangle_{H^2}\quad(f\in K_\Theta), \] where the inner product is linear in its first argument. For nonconstant $\Theta$, put \[ e_\lambda^\Theta=\frac{k_\lambda^\Theta}{\|k_\lambda^\Theta\|}, \qquad \|k_\lambda^\Theta\|^2=\frac{1-|\Theta(\lambda)|^2}{1-|\lambda|^2}. \] An unconditional basis is a family such that every vector has a unique expansion in that family, with convergence in norm independent of the order of summation.

Question under the stated convention. Is it true that for every nonconstant inner function $\Theta$ there exist a finite or countably infinite index set $J$ and distinct points $\lambda_j\in\mathbb D$, $j\in J$, such that $(e_{\lambda_j}^\Theta)_{j\in J}$ is an unconditional basis of $K_\Theta$? Equivalently, is there such a family with dense linear span in $K_\Theta$ and constants $0<A\leq B<\infty$ satisfying \[ A\sum_{j\in J}|c_j|^2 \leq \left\|\sum_{j\in J}c_j e_{\lambda_j}^\Theta\right\|_{H^2}^2 \leq B\sum_{j\in J}|c_j|^2 \] for every finitely supported complex family $(c_j)_{j\in J}$? The points and constants may depend on $\Theta$. Normalization does not change the unconditional-basis question for nonzero kernels. For a constant inner function, $K_\Theta=\{0\}$, which may be handled by the empty-basis convention.

The source does not establish whether kernels at boundary points admitting bounded evaluation are also allowed, or whether another Hardy-space domain is intended. These choices must be checked before this conditional formulation is accepted as the intended statement.

Short English statement

The supplied problem asks whether every model space $H^2\ominus\Theta H^2$ has an unconditional basis consisting of reproducing kernels. The displayed formulation uses the unit disc and interior kernels explicitly; the source record leaves the domain and admissible kernel locations unresolved. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Supplied local-registry record, permanent UnsolvedMath ID $30001609$, source problem code catalogue entry, title: Unconditional Reproducing-Kernel Bases in Model Spaces. Project registry references: data/unsolvedmath/problems.json and catalog/problems.json; no independent file inspection was performed.
• Supplied source citation: Operator Theory and Harmonic Analysis (2011), Oberwolfach Reports, pp. 2813--2875, DOI 10.4171/owr/2010/49. Supplied source URL and literature-source link labelled "original problem report": https://doi.org/10.4171/owr/2010/49. No individual proposer is supplied. The record gives the proposed year as 2010.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says that the report explicitly presents the general existence question as open, and reports that later work provides criteria and examples without a general resolution being found. This is supplied dated curation, not independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, accompanied by a report of manual semantic, typographic, and LaTeX review. This supplied editorial label does not independently establish the intended domain, boundary-kernel convention, or mathematical status.