An operators representation for weighted inductive limits of spaces of vector valued holomorphic functions. (Q1812201)

Let \({\mathcal V}= (v_n)_n\) be a decreasing sequence of (strictly) positive functions on an open set \(G\subset\mathbb{C}^N\). For a quasicomplete locally convex space \(E\), the weighted inductive limit of spaces of vector valued holomorphic functions is defined by \({\mathcal V}H(G, E):= \text{ind}_n Hv_n(G, E)\) (\({\mathcal V}H(G)\) is the corresponding space of scalar functions). The ``projective hull'' of \({\mathcal V}H(G, E)\) is denoted by \(H\overline V(G, E)\), where \(\overline V= \{\overline v> 0\) continuous on \(G: \sup_G\overline v/v_n <\infty\,\forall n\in\mathbb{N}\}\) is the Nachbin family. If the two spaces are equal algebraically and topologically, i.e., \({\mathcal V}H(G, E)\simeq H\overline V(G, E)\), one says that the \(E\)-valued projective description holds. In the paper under review, the problem when the operator representation \[ {\mathcal V}H(G,E_b')= \text{ind}_n{\mathcal L}_b(E, Hv_n(G))={\mathcal L}_b(E,{\mathcal V}H(G)) \] holds algebraically and topologically (i.e., when the inductive limit \(\text{ind}_n Hv_n(G)\) interchanges with the space of continuous linear operators defined on \(E\)), is studied. The algebraic equality is valid under very general conditions, e.g., whenever \(E\) is a Fréchet space. The authors show that in the case that \(E\) is a Fréchet space with the density condition and \({\mathcal V}H(G)\) is regularly decreasing, the topological equality \({\mathcal V}H(G, E_b')= {\mathcal L}_b(E, {\mathcal V}H(G))\) holds if and only if the pair \((E, F)\) of Fréchet spaces has the property (BB) (i.e., if Grothendieck's ``problème des topologies'' has a positive solution for \(E\widehat\otimes_\pi F\)), where \(F\) is the predual of \({\mathcal V}H(G)\). The authors point out that, in terms of the conditions of \textit{A. Peris} [Ann. Acad. Sci. Fenn., Ser. AI Math. 19, 167--203 (1994; Zbl 0789.46006)], the desired topological equality holds if \({\mathcal V}H(G)\) is (QNo), i.e., it satisfies the strict Mackey condition by operators, and \(E\) is (QNo), quasinormable by operators, which is true if, e.g., \(E\) is a Banach space. Such locally convex properties by operators are related to Grothendieck's ``problème des topologies'', which had been solved in the negative by \textit{J. Taskinen} [Counterexamples to ``Problème des topologies'' of Grothendieck, Ann. Acad. Sci. Fenn., Ser. AI, Diss. 63 (1986; Zbl 0612.46069)]. It is also demonstrated that the problem of operator representation considered in this paper is connected with the question when the vector valued description holds as follows: If the projective description holds in the scalar case, then \(H\overline V(G,E_b')= {\mathcal L}_b(E, {\mathcal V}H(G))\) and hence \({\mathcal V}H(G, E_b')= G_b(E, {\mathcal V}H(G))\) is equivalent to \({\mathcal V}H(G, E_b')= H\overline V(G, E_b')\).