Research Problems in Function Theory — Problem 8.10

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. For $p>0$, the Bergman space is $A^p(\mathbb D)=\{f\text{ analytic}:\|f\|_{A^p}=(\int_{\mathbb D}|f|^p\,dA)^{1/p}<\infty\}$, where $dA=dx\,dy$ is ordinary area measure. For $f\in A^2$ write $\mathcal Pf=\{pf:p\in\mathbb C[z]\}$. Call $f$ cyclic if the closure of $\mathcal Pf$ in the $A^2$ norm is $A^2$.

Statement recorded in the supplied source.

Is every $f$ for which both $f$ and its analytic reciprocal $1/f$ belong to $A^2$ cyclic? More generally, if $f\in A^2$ and there are $a,c>0$ with $|f(z)|\ge c(1-|z|)^a$ for every $z\in\mathbb D$, must $f$ be cyclic? [S2]

Short English statement

Do an integrable analytic reciprocal or the stated polynomial lower bound guarantee Bergman cyclicity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308010 (AMR-022-8010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 8.10, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.