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Homomorphisms on Hadamard algebras. (English) Zbl 0884.46035

Marino, G. (ed.) et al., The proceedings of the workshop on functional analysis: methods and applications, Camigliatello Silano, Italy, May 29–June 2, 1994. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 40, 153-158 (1996).
Summary: The aim of this paper is to study homomorphisms on the algebra \(H(G)\) of all holomorphic functions on \(G\) endowed with the Hadamard product. It is shown that each homomorphism is continuous. For a homomorphism \(\Phi: H(G_1)\to H(G_2)\) there exists a bijective mapping \(\phi:\mathbb{N}_0\to \mathbb{N}_0\) such that for each \(f\in H(G_1)\) the function \(\Phi(f)\) is locally given by \(\sum^\infty_{n= 0}a_nz^{\phi(n)}\), where \(f(z)= \sum^\infty_{n= 0}a_nz^n\). Two domains \(G_1\) and \(G_2\) are called Hadamard-isomorphic if \(H(G_1)\) and \(H(G_2)\) are isomorphic (with respect to the Hadamard product). It is proved that the open unit disk \(D\) is not Hadamard-isomorphic to any other admissible domain. Moreover, simply connectedness is an invariant of (Hadamard) isomorphisms.
For the entire collection see [Zbl 0847.00038].

MSC:

46J05 General theory of commutative topological algebras
46H40 Automatic continuity
30A99 General properties of functions of one complex variable
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