Closed submodules of holomorphic functions with two generators. (English. Russian original) Zbl 1078.30047

Khabibullin, B. N.

Funct. Anal. Appl. 38, No. 1, 52-64 (2004); translation from Funkts. Anal. Prilozh. 38, No. 1, 65-80 (2004).

For a planar domain \(\Omega\), let \(H(\Omega)\) be the set of all holomorphic functions in \(\Omega\) and let \({\mathcal P}\subseteq H(\Omega)\) be a separable, locally convex vector space of holomorphic functions. It is assumed that convergence in \(\mathcal P\) implies pointwise convergence in \(H(\Omega)\) and that multiplication by the independent variable \(z\) is a continuous operation on \(\mathcal P\). The aim of the paper is to give sufficient conditions on a \(z\)-invariant, closed subspace of \(\mathcal P\) to be topologically two generated. The conditions, though, are too technical to be citated here. The paper generalizes research done by the author in [Mat. Zametki 76, No. 4, 604–609 (2004); translation in Math. Notes 76, No. 4, 558–563 (2004; Zbl 1068.30025)], for ideals in subalgebras of \(H(\Omega)\).

Reviewer: Raymond Mortini (Metz)

MSC:

30H05	Spaces of bounded analytic functions of one complex variable
46E10	Topological linear spaces of continuous, differentiable or analytic functions

Keywords:

\(z\)-invariant subspaces; locally convex subspaces of holomorphic functions

Citations:

Zbl 1068.30025