Research Problems in Function Theory — Problem 7.59

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.

For $n\ge1$ let $\mathcal E=\mathcal O(\mathbb C^n)$ be all entire functions. For a multi-index $\alpha=(\alpha_1,\ldots,\alpha_n)\in\{0,1,\ldots\}^n$, put $z^\alpha=\prod z_j^{\alpha_j}$ and $D^\alpha=\prod(\partial/\partial z_j)^{\alpha_j}$. If $P(z)=\sum_\alpha p_\alpha z^\alpha$ is a polynomial, define $P(D)=\sum_\alpha p_\alpha D^\alpha$ and $T_{P,Q}f=P(D)(Qf)$. Entire exponential type means $|f(z)|\le A e^{B|z|}$ for some finite positive $A,B$.

Statement recorded in the supplied source.

For arbitrary complex polynomials $P,Q$ in $n$ variables, does bijectivity of $T_{P,Q}:\mathcal E\to\mathcal E$ imply bijectivity of $T_{Q,P}:\mathcal E\to\mathcal E$? The question concerns all entire functions, not only those of exponential type. The source records that the reversed operator is bijective on the exponential-type class under the hypothesis, but its update gives a negative answer on the full entire class. [S2]

Short English statement

Historical question: can one interchange the multiplication polynomial and the constant-coefficient differential polynomial while preserving bijectivity on all entire functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307059 (AMR-022-7059). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.59, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.