On the property (LB\(^\infty\)) of spaces of germs of holomorphic functions and the properties (\(\tilde\Omega \bar\Omega\)) of the Hartogs domains in infinite dimension (Q1577610)

Let \(E\) be a Fréchet space with an absolute basis. For a balanced convex compact subset \(K\) of \(E\) denote by \(H(K)\) the Fréchet space of all germs of holomorphic functions on \(K\). Using the given basis, the authors associate with \(K\) a decreasing family \((K_\varepsilon)_{\varepsilon> 0}\) of balanced convex compact sets and prove that the following assertions are equivalent: (1) \(E\) has the property \((\widetilde\Omega_{k_\varepsilon})\) for some \(/\) all \(\varepsilon> 0\). (2) \(H(K_\varepsilon)'\) has the property \((LB^\infty)\) for some \(/\) all \(\varepsilon> 0\). (3) The set \(K_\varepsilon\) is not pluripolar in \(E\) for some \(/\) all \(\varepsilon> 0\). The equivalence of (1)--(3) for arbitrary balanced convex compact subsets of nuclear Fréchet spaces with bounded approximation property was shown by \textit{Nguyen Dinh Lan} [Publ. Mat., Barc. 44, No. 1, 177-192 (2000)], using earlier results of \textit{D. Vogt} [J. Reine Angew. Math. 345, 182-200 (1983; Zbl 0261.46003)] who characterized the property \((LB^\infty)\) and \textit{S. Dineen}, \textit{R. Meise} and \textit{D. Vogt} [Bull. Soc. Math. Fr. 112, 41-68 (1984; Zbl 0556.46003)] who proved the equivalence of (1) and (3) under the latter hypotheses. In the second part of the paper the authors want to extend a result of \textit{A. Aytuna} [Manuscr. Math. 62, 297-315 (1988; Zbl 0662.32014)] who proved that the Fréchet space \(H(\Omega_\varphi)\) of all holomorphic functions on the Hartogs domain \(\Omega_\varphi\) in \(\mathbb{C}^{n+1}\) over the pseudoconvex domain \(\Omega\) in \(\mathbb{C}^n\) has the property \((\overline\Omega)\) whenever \(H(\Omega)\) has \((\overline\Omega)\). However, the authors seem not to notice that even for open polydiscs \(\mathbb{D}_a\) in a nuclear Fréchet space with basis, the space \(H(D_a)\) of all holomorphic on \(\mathbb{D}_a\) is not a Fréchet space.