Abstract

The system \(\mathcal D_0\) of partial backward shift operators in a countable inductive limit \(E\) of weighted Banach spaces of entire functions of several complex variables is studied. Its commutant \(\mathcal K(\mathcal D_0)\) in the algebra of all continuous linear operators on \(E\) operators is described. In the topological dual of \(E\), a multiplication \(\circledast\) is introduced and studied, which is determined by shifts associated with the system \(\mathcal D_0\). For a domain \(\Omega\) in \(\mathbb C^N\) polystar-shaped with respect to 0, Duhamel product in the space \(H(\Omega)\) of all holomorphic functions on \(\Omega\) is studied. In the case where, in addition, the domain \(\Omega\) is convex, it is shown that the operation \(\circledast\) is realized by means of the adjoint of the Laplace transform as Duhamel product.

中文翻译：

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全纯函数空间中的多维杜哈梅尔积和后移算子

摘要

研究了多复变量全函数加权Banach空间的可数归纳极限\(E\)内的部分后移算子系统\(\mathcal D_0\) 。描述了其在所有连续线性算子代数中对\(E\)算子的交换律\(\mathcal K(\mathcal D_0)\) 。在\(E\)的拓扑对偶中，引入并研究了乘法\(\circledast\)，它由与系统\(\mathcal D_0\)相关的移位确定。对于\(\mathbb C^N\)中相对于 0 呈多星形的域\(\Omega\ )，空间中的杜哈梅尔积研究了\(\Omega\)上所有全纯函数的\ (H (\Omega)\) 。另外，在域\(\Omega\)是凸的情况下，表明运算\(\circledast\)是通过作为杜哈梅尔乘积的拉普拉斯变换的伴随来实现的。