The density condition and distinguishedness for weighted Fréchet spaces of holomorphic func\-tions (Q868801)

The author deals with Fréchet spaces of the following type: Let \(G\subset\mathbb{C}^N\) be open and balanced, let \(W\) be an increasing sequence of radial weights and let \(HW(G)\) and \(HW_0(G)\) denote the corresponding weighted Fréchet spaces of holomorphic functions (of type \(H^\infty\)). She investigates the properties ``density condition'' (DC) and distinguishedness for Fréchet spaces of that type. She first shows that a certain condition on the weights is necessary for \(HW(G)\) to have DC if \(HW(G)\) contains all polynomials. Next she proves that under additional assumptions (e.g., the weights are taken from a special class, \(HW_0(G)\) contains all polynomials, etc.) this condition is also sufficient. Finally, utilizing the result that there exist nondistinguished Fréchet spaces of Köthe echelon type, she constructs a Fréchet space \(HW(G)\) which is not distinguished.