Abstract
Several representation theorems of multipliers are derived: in terms of analytic functionals, and germs of holomorphic functions. The co-induced topology via the representation theorems is discussed. As an application of the representation theorems the density of non-invertible multipliers is proved. Moreover, Euler differential operators are distinguished among all multipliers.
Highlights
Several representation theorems of multipliers are derived: in terms of analytic functionals, and germs of holomorphic functions
As Domanski and Langenbruch we give the desciption of all Hadamard multipliers on the space H(Ω) via analytic functionals H(V (Ω)) on the space of holomorphic germs on the dilation set of Ω, i.e., V (Ω) := {z ∈ C : zΩ ⊂ Ω}
The second problem that we look at is related to the co-induced topology on H(V ) by Φ from M(Ω), i.e., the topology on H(V ) that makes Φ defined in Theorem 1.1 a topological isomorphism
Summary
Following Hadamard define the Hadamard multiplication on the space of germs of holomorphic functions around the origin:. As Domanski and Langenbruch we give the desciption of all Hadamard multipliers on the space H(Ω) via analytic functionals H(V (Ω)) on the space of holomorphic germs on the dilation set of Ω, i.e.,. The celebrated Hadamard’s Theorem provides the answer to the question concerning the domain of existence of the Hadamard product of Taylor series around the origin (for the classical formulation cf [4, Chapter 1.4], or [19, Chapter 6.3]; for the extended one see [18]) It says nothing about the possible holomorphic extension of f ∗ g beyond its star product. Where Mφ h := h ∗ φ, Ω ⊂ C is a Runge open set containing the origin, and ∗ denotes the Hadamard product. For non-explained notions from Functional Analysis we refer to the book [15]
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