Linear topological invariants and splitting of closed ideals in weighted algebras of analytic functions on the disc
This paper deals with a class of locally convex subalgebras of A(\({\mathbb{D}})\), the algebra of holomorphic functions on the open unit disc \({\mathbb{D}}\). The main objective is to determine when an algebra in the class has the property that every closed ideal in the algebra is (topologically linearly) complemented. By a weight system (or inductive weight system) is meant a family \(\Psi =(\psi_ k)_{k\in {\mathbb{N}}}\) of continuous, nonnegative, strictly increasing, unbounded functions defined for \(x\geq 1\) and satisfying four growth and monotonicity (relative to the subscript k as well as the variable x) conditions. Associated to \(\Psi\) is a system \({\mathbb{P}}=(p_ k)_{k\in {\mathbb{N}}}\) of functions on \({\mathbb{D}}\) given by \(p_ k(z)=\psi_ k((1-| z|)^{-1})\). Then \(A_{{\mathbb{P}}}({\mathbb{D}})\), the algebra under study, is the space of functions f in A(\({\mathbb{D}})\) such that \(\| f\|_ k\equiv \sup \{| f(z)| e^{-p_ k(z)}:\;z\in {\mathbb{D}}\}\) is finite for some \(k\in {\mathbb{N}}\), in the topology induced by the \(\| \cdot \|_ k.\) Theorem 1.11 gives four characterizations of when \(A_{{\mathbb{P}}}({\mathbb{D}})\) has the complementation property mentioned above. The one which can be stated directly in terms of \(\Psi\) is this: ``There exists \(m\in {\mathbb{N}}\) such that for each \(k\in {\mathbb{N}}\) there exists \(n\in {\mathbb{N}}\) and \(x_ 0\geq 1\) with \(\psi_ k(x)\psi_ n^{- 1} (\psi_ k(x))\leq x\psi_ m(x)\), \(x\geq x_ 0.'' \) A parallel result, Theorem 2.5, treats the case when \(\Psi\) is a projective weight system (similar to ``inductive weight system'', but various quantifiers and the order of some inequalities are changed), and \(\| f\|_ k\) is finite for all (not just some) \(k\in {\mathbb{N}}\). The author refers frequently to a preprint of his, which may not be accessible to many readers. He also observes that various cases have previously been investigated in which instead of \({\mathbb{D}}\), other plane regions - notably \({\mathbb{C}}\)- have been used. In particular, an important article by \textit{R. Meise} and \textit{B. A. Taylor} [Trans. Am. Math. Soc. 302, 341-370 (1987; Zbl 0621.32022)], which actually looks at \({\mathbb{C}}^ n\), is cited.
algebra of holomorphic functions on the open unit disc
scientific article; zbMATH DE number 4210849
scientific article