All closed ideals, of the algebra $A_\varphi(\mathbb C)$ are divisorial

S. A. Apresyan

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 253-258

Résumé

Let $\lambda$ be an increasing function on the half-line (satis fying some regularity growth conditions), $A_\lambda$ the algebra of all entire functions $f$ satisfying $\log|f(z)|=0(\lambda(|z|))$ ($|z|\to\infty$). It is proved that every closed ideal $I$ of the algebra $A_\lambda$ is divisorial, i.e.  $I=I_k\overset{\text{def}}=\{f\in A_\lambda:k_f(\xi)\ge k_I(\xi),\xi\in\mathbb C\}$, $k_f(\xi)$ being the multiplicity of the zero of $f$ at $\xi$, $k_I(\xi)=\min_{f\in I}k_f(\xi)$, $\xi\in\mathbb C$. It is shown that $f\equiv0$ provided $f\in A_\lambda$, $$ \lim_{\substack{\xi\in\gamma\\|\xi|\to\infty}}\frac{\log|f(\xi)|}{\lambda(|\xi|)}=-\infty $$ where $\gamma$ denotes continuous curve joining the origin with the infinity.