Wandering subspaces of the Bergman space and the Dirichlet space over polydisc

Abstract

Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc ${\mathbb{D}}^n$ (with $n\ge 2$) is investigated. We show that for any non-empty subset $\alpha =\{{\alpha }_1,\dots ,{\alpha }_k\}$ of $\{1,\dots ,n\}$ and doubly commuting invariant subspace $\s$ of the Bergman space or the Dirichlet space over $\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_α|_\s:=(M_{z_{α_1}}|_\s,\dots, M_{z_{α_k}}|_\s)$ always possesses wandering subspace of the form \[\bigcap_{i=1}^k(\s\ominus z_{α_i}\s). \]

arXiv comment10 pages

Cited by in corpus (1)

• Rudin's Submodules of $H^2({\mathbb{D}}^2)$