Projective limits of weighted LB-spaces of holomorphic functions (Q2275503)

The paper under review deals with projective limits of weighted LB-spaces of holomorphic functions. It is divided into seven parts. The introduction gives brief information on the theory of PLB-spaces together with pointing out how homological algebra methods may be applied to functional analytic questions. Section 2 recalls the definition of Banach spaces of holomorphic functions \(Ha(G)\) and \(Ha_0(G)\), where \(G\) is an open subset in \(\mathbb{C}^d\) and \(a\) is a weight on \(G\). Next, the PLB-spaces of holomorphic functions are recalled. Later, the author defines several versions of the conditions (Q) and (wQ) introduced in [\textit{D. Vogt}, Progress in functional analysis, Proc. Int. Meet. Occas. 60th Birthd. M. Valdivia, Peníscola/Spain, North-Holland Math. Stud. 170, 57--84 (1992; Zbl 0779.46005)] which he uses in the sequel. Section 3 deals with barrelledness and ultrabornologicity of \(AH(G)\) and \(AH_0(G)\), and under the additional assumptions that the domain be balanced and the weights radial, he obtains necessary conditions. The next section gives sufficient conditions for the above mentioned properties. Here we need some stronger assumptions on the weights -- roughly speaking, they must be of class \(\mathcal{W}\) defined by \textit{K. D. Bierstedt} and \textit{J. Bonet} [Proc. Edinb. Math. Soc., II. Ser. 46, No. 2, 435--450 (2003; Zbl 1060.46018)]. In Theorems 3.1, 3.5, 4.1 and 4.3, the conditions of type (Q) of the author appear. Section 5 shows when a PLB-space \(AH(G)\) or \(AH_0(G)\) can be viewed as a weighted LF-space of holomorphic functions. However, in the special case of the unit disc it is possible to give equivalent conditions for \(AH(\mathbb{D})\) and \(AH_0(\mathbb{D})\) to be barrelled and ultrabornological. This is the content of Section 6. The last section gives examples of weights \(A\) when \(AH(\mathbb{D})\) and \(AH_0(\mathbb{D})\) are proper PLB-spaces.