Research Problems in Function Theory — Problem 5.26

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$.The Bergman space $A^2(\mathbb D)$ is the Hilbert space of holomorphic functions with \[\|f\|_{A^2}^2=\frac1\pi\int_{\mathbb D}|f(z)|^2\,dA(z)=\sum_{n=0}^\infty\frac{|a_n|^2}{n+1}<\infty, \] where $dA$ is planar area. The condition is equivalent to $\sum_{n\ge1}|a_n|^2/n<\infty$ used in the source. A linear subspace $L\subset A^2(\mathbb D)$ is invariant under the Bergman shift $M_z$ if $zf\in L$ for every $f\in L$. A closed subspace means closed for this Hilbert norm.

Statement recorded in the supplied source.

Describe the invariant subspaces of $A^2(\mathbb D)$, in particular classify all its closed $M_z$-invariant subspaces. Give a necessary and sufficient structural description, analogous in purpose to the Hardy-space invariant-subspace theorem. The zero subspace and the whole space are included. [S2]

Short English statement

Classify the invariant subspaces of the Bergman shift, including a structural description of every closed invariant subspace.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305026 (AMR-022-5026). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.26, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.