Holomorphic Cohomological Convolution and Hadamard Product

Dubussy, Christophe; Schneiders, Jean-Pierre

doi:10.4171/prims/58-1-2

Publ. Res. Inst. Math. Sci.

2022

DOI: 10.4171/prims/58-1-2

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Holomorphic Cohomological Convolution and Hadamard Product

Christophe Dubussy

Jean-Pierre Schneiders²

Abstract: In this article we explain the link between Pohlen's extended Hadamard product and the holomorphic cohomological convolution on C * . For this purpose we introduce a generalized Hadamard product, which is defined even if the holomorphic functions do not vanish at infinity, as well as a notion of strongly convolvable sets.

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Multipliers on Spaces of Holomorphic Functions

Trybuła

2023

Complex Anal. Oper. Theory

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We consider multipliers on the space of holomorphic functions of one variable $$H(\Omega ),\,\Omega \subset \mathbb {C}$$ H ( Ω ) , Ω ⊂ C open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to $$\Omega $$ Ω and $$\Omega $$ Ω is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on $$H(\Omega )$$ H ( Ω ) . We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when $$\Omega $$ Ω is convex.

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Multipliers on Spaces of Holomorphic Functions

Trybuła

2023

Complex Anal. Oper. Theory

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Holomorphic Cohomological Convolution and Hadamard Product

Cited by 1 publication

References 9 publications

Multipliers on Spaces of Holomorphic Functions

Multipliers on Spaces of Holomorphic Functions

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