The Hadamard product in the space of Lorch analytic mappings
Given a complex Banach algebra \(E,\) a self-map \(f\) of \(E\) is said to be Lorch analytic if, for every \(z\in E,\) there is an \(a(z)\in E\) such that the Fréchet derivative of \(f\) at \(z\) is of the form \(df_z(h)=ha(z)\), \(h\in E\). This notion was introduced by \textit{E. R. Lorch} [Trans. Am. Math. Soc. 54, 414--425 (1943; Zbl 0060.27202)]. The nice feature of \(f\in H_L(E)\) is that its Taylor series at \(0\) is of the form \(f(z)=\sum_{n=0}^\infty a_nz^n\) for some sequence \((a_n)_n\subset E\) satisfying \(\lim_n \|a_n\|^{1/n}=0.\) This allows the authors to define a Hadamard product on \(H_L(E)\) according to \((f\cdot g)(z):=\sum_{n=0}^\infty a_nb_nz^n\) for \(g=\sum_{n=0}^\infty b_nz^n\in H_L(E)\). The authors consider \(H_L(E)\) endowed with the pointwise sum and scalar product, the Hadamard product and the topology of uniform convergence on balls, which they prove is a commutative \(m\)-convex Fréchet algebra. The first section of the paper also deals with \(\mathcal{M}(H_L(E)),\) the spectrum of this algebra, that is shown to be homeomorphic to \(\mathcal{M}(E)\times (\mathbb{N}\cup \{0\})\) (here, spectra are endowed with the Gelfand topology) and it is also shown that \(H_L(E)\) is semi-simple if \(E\) is. In the second section, the authors deal with the sequence space \(\Gamma(E):=\{(a_n)_n\subset E \text{ such that } \lim_n \|a_n\|^{1/n}=0\}\) with the metric defined by \[ d((a_n),(b_n)):=\sup\{\|a_0-b_0\|,\|a_n-b_n\|^{1/n}:n\in\mathbb{N}\}. \] It is proved that \((\Gamma(E),d)\) is a complete topological vector space isomorphic to \(H_L(E),\) hence a locally convex one. With the usual coordinate-wise product, \((\Gamma(E),d)\) is a commutative Fréchet algebra about whose spectrum some consequences are derived from the mentioned isomorphism.
scientific article; zbMATH DE number 6153053
scientific article