Geodesic
Many-Dimensional Duhamel Product in the Space of Holomorphic Functions and Backward Shift Operators
Matematičeskie zametki, Tome 113 (2023) no. 5, pp. 677-692

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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 commutator subgroup $\mathcal K(\mathcal D_0)$ 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.
Keywords: Duhamel product, backward shift operator, space of holomorphic functions. 
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     author = {P. A. Ivanov and S. N. Melikhov},
     title = {Many-Dimensional {Duhamel} {Product} in the {Space} of {Holomorphic} {Functions} and {Backward} {Shift} {Operators}},
     journal = {Matemati\v{c}eskie zametki},
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P. A. Ivanov; S. N. Melikhov. Many-Dimensional Duhamel Product in the Space of Holomorphic Functions and Backward Shift Operators. Matematičeskie zametki, Tome 113 (2023) no. 5, pp. 677-692. http://geodesic.mathdoc.fr/item/MZM_2023_113_5_a4/