Research Problems in Function Theory — Problem 7.61

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 $A,B$.Define $P^*(z)=\overline{P(\bar z)}=\sum_\alpha\overline{p_\alpha}z^\alpha$. The zero polynomial makes $T_{P^*,P}$ the zero operator; the substantive bijection question is for nonzero $P$, including nonzero constants.

Statement recorded in the supplied source.

The original asks whether $T_{P^*,P}:\mathcal E\to\mathcal E$ is bijective for every polynomial $P$. For $P=0$ the answer is negative, since the operator is zero. For every nonzero complex polynomial $P$, is this operator bijective? Also consider the source's proposed sufficient route: for every multi-index $\alpha$, does \[P^*(D)(Pf)=z^\alpha \] admit an entire solution of exponential type? The source records the homogeneous case as known and notes the injectivity claim separately; neither replaces the full nonhomogeneous bijection question. [S2]

Short English statement

Determine bijectivity of polynomial multiplication followed by the differential operator with conjugate coefficients, and whether exponential-type solutions exist for every monomial right-hand side.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307061 (AMR-022-7061). 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 7.61, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.