A Differential Operator Representation of Continuous Homomorphisms Between the Spaces of Entire Functions of Given Proxi
- PDF / 396,007 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 42 Downloads / 243 Views
Complex Analysis and Operator Theory
A Differential Operator Representation of Continuous Homomorphisms Between the Spaces of Entire Functions of Given Proximate Orders Takashi Aoki1
· Ryuichi Ishimura2 · Yasunori Okada3
Received: 27 March 2020 / Accepted: 29 August 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we consider the locally convex spaces of entire functions with growth given by proximate orders, and study the representation as a differential operator of a continuous homomorphism from such a space to another one. As a corollary, we give a characterization of continuous endomorphisms of such spaces. Keywords Entire functions · Proximate orders · Partial differential operators of infinite order Mathematics Subject Classification Primary 34A15 · Secondary 47F99
This article is part of the topical collection “In memory of Carlos A. Berenstein (1944–2019)” edited by Irene Sabadini and Daniele Struppa. Communicated by Irene Sabadini. The first author is supported by JSPS KAKENHI Grant Nos. 26400126 and 18K03385. The third author is supported by JSPS KAKENHI Grant No. 16K05170.
B
Takashi Aoki [email protected] Ryuichi Ishimura [email protected] Yasunori Okada [email protected]
1
Department of Mathematics, Kindai University, Higashi¯osaka 577-8502, Japan
2
Department of Mathematics and Informatics, Faculty of Science, Chiba University, Yayoicho, Chiba 263-8522, Japan
3
Institute of Management and Information Technologies, Chiba University, Yayoicho, Chiba 263-8522, Japan 0123456789().: V,-vol
75
Page 2 of 22
T. Aoki et al.
1 Introduction A linear differential operator often defines a linear continuous operator on a function space when the considering space is endowed with a natural topology, and sometimes also defines an endomorphism of a sheaf of functions. Therefore, there arises a natural question whether every linear continuous operator or every endomorphism of a given space or sheaf of functions is represented as a differential operator, and this problem has been studied in various situations. In the smooth cases, Jaak Peetre [16] and [17] proved that an endomorphism of the sheaf of smooth functions is noting but a linear differential operator locally of finite order. In the analytic cases or ultradifferentiable cases, a continuous endomorphism of corresponding sheaf is characterized rather as a differential operator of infinite order with symbol satisfying certain growth condition ([18] and [12]). For the cases of entire functions, in the paper [3], we proved with coauthors Struppa and Uchida that any continuous endomorphism of the space of entire functions with a given constant order ρ > 0 is characterized as an infinite order partial differential operator with the symbol satisfying certain growth conditions. As remarked in [3], we can largely generalize these results to the case of proximate order in the sense of Valiron [19], instead of a constant order. Such generalizations were first proved by Jin [11, Corollaries 6.5 and 6.6], where he c
Data Loading...
