Bergman spaces on disconnected domains

Summary, in English

For a bounded open set $U$ in the complex plane, we consider the Bergman space $L^p_a(U)$ of $p$th-power area-integrable analytic functions on $U$, with $p$ assumed to be in the range $1\le p<+\infty$, where the lower bound assures that $L^p_a(U)$ is a Banach space. Fix a bounded domain $G$ and a compact subset $K$ of $G$ with zero area measure. Let $M$ be a closed subspace of $L^p_a(G\sbs K)$. We say that $M$ is invariant provided that $zf\in M$ whenever $f\in M$. Aleman, Richter, and Ross study those invariant subspaces $M$ of $L^p_a(G\sbs K)$ that contain $L^p_a(G)$. They also add the requirement that $M$ have index one, which is taken to mean that $(z-\lambda)M$ has codimension $1$ in $M$, for all $\lambda\in G\sbs K$. Natural examples of such invariant subspaces are those of the form $M=L^p_a(G\sbs E)$, where $E$ is a closed subset of $K$. The authors show that for $p<2$, these are indeed all such invariant subspaces which can be found. For $p\ge2$, this is not so, but nevertheless a complete classification can be found in terms of quasi-closed subsets $E$ of $K$.



If the condition that the index of $M$ is one is dropped, then the structure of such invariant subspaces can be extremely complicated. If, however, $G\sbs K$ is connected, the authors show that $M$ automatically has index one (due to the assumption that $M$ should contain $L_a^p(G)$).

• Författare: Alexandru Aleman; Stefan Richter; William T Ross
• Publiceringsår: 1996
• Språk: Engelska
• Sidor: 225-243
• Publikation/Tidskrift/Serie: Canadian Journal of Mathematics
• Volym: 48
• Issue: 2
• Dokumenttyp: Artikel i tidskrift
• Förlag: Canadian Mathematical Society
• Ämne: Mathematics
• Status: Published
• ISBN/ISSN/Övrigt: ISSN: 0008-414X