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.
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 677–692 https://doi.org/10.4213/mzm13755.
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Ivanov, P.A., Melikhov, S.N. Many-Dimensional Duhamel Product in the Space of Holomorphic Functions and Backward Shift Operators. Math Notes 113, 650–662 (2023). https://doi.org/10.1134/S000143462305005X
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DOI: https://doi.org/10.1134/S000143462305005X