Abstract
This paper mainly considers cyclic vectors in the Fock-type spaces $ L_{a,\alpha}^{p,s }(\mathbb{C} )$ $(\alpha>0,p\geq 1, s>0)$ which consists of all entire functions $f$ such that $|f|^p$ is integrable with respect to the measure $\exp(-\alpha |z|^s) dA(z).$ The case of $s$ not being an integer was done in [9], where cyclic vectors are exactly those non-vanishing entire functions in $ L_{a,\alpha}^{p,s }(\mathbb{C} )$. In this paper it is shown that for each positive integer $s$, a function $f$ is cyclic in $ L_{a,\alpha}^{p,s }(\mathbb{C} )$ if and only if $f$ is non-vanishing and $f \mathcal{C} \subseteq L_{a,\alpha}^{p,s }(\mathbb{C} )$, where $ \mathcal{C} $ denotes the polynomial ring. Moreover, the condition that $f \mathcal{C} \subseteq L_{a,\alpha}^{p,s }(\mathbb{C})$ can not be dropped.
Acknowledgment
The authors are quite grateful to the referee for many valuable suggestions that make this paper more readable.
Citation
Hansong Huang. Kou Hei Izuchi. "Cyclic vectors in Fock-type spaces of single variable." Nihonkai Math. J. 28 (2) 117 - 124, 2017.
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