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arXiv · 1805.00663

Characterization of continuous endomorphisms in the space of entire functions of a given order

Takashi Aoki·Ryuichi Ishimura·Yasunori Okada·Daniele C. Struppa·Shofu Uchida
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Abstract

The aim of this paper is to characterize continuous endomorphisms in the space of entire functions of exponential type of order $p>0$. Let $A_p$ denote the space of entire functions of $n$ complex variables $z\in{\mathbb C}^n$ of order $p$ of normal type. We consider an endomorphism $F$ in the space, which is considered to be a DFS-space. We show that there is a unique linear differential operator $P$ of infinite order with coefficients in the space which realizes $F$, that is, $Ff=Pf$ holds for any $f\in A_p$. The coefficients satisfy certain growth conditions and conversely, if a formal differential operator of infinite order with coefficients in $A_p$ satisfy these conditions, then it induces a continuous endomorphism.

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Takashi Aoki, Ryuichi Ishimura, Yasunori Okada, Daniele C. Struppa, Shofu Uchida. 2018-05-02. Characterization of continuous endomorphisms in the space of entire functions of a given order. https://doi.org/10.1080/17476933.2020.1767086

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