The problem of generators in certain rings of analytic functions (Q1893645)

Let \(H_\psi (D)\) be the weight space of analytic functions in the domain \(D \subset \mathbb{C}^n\), where the weight \(\psi (z)\) is a nonnegative plurisubharmonic function which satisfies some conditions. The author denotes by \(H^p_\psi\), \(p \geq 1\), the following set \[ \begin{multlined} H^p_\psi = \Bigl \{f : f(z) = \bigl( f_1(z), \ldots, f_m (z), \ldots \bigr),\;f_j (z) \in H_\psi (D),\;\forall j, \\ \sum^\infty_{j = 1} \bigl |f_j (z) \bigr |^p \leq A_1 \exp \bigr\{ A_2 \psi (z) \bigr\},\;z \in D \Bigr\}. \end{multlined} \] For any element \(f(z) \in H^p_\psi\), \(p \geq 2\), \[ I(f) : = \left\{ \sum^\infty_{j = 1} f_j (z) g_j (z),\;g(z) \in H^q_\psi,\;1/p + 1/q = 1 \right\}. \] It is clear that \(I(f) \subset H_\psi (D)\) for \(f \in H^p_\psi\), \(p \geq 2\). The author obtains necessary and sufficient conditions for the equality \(I(f) = H_\psi (D)\).