Revistes Catalanes amb Accés Obert (RACO)

Invariant subspaces on multiply connected domains

A. Abkar, H. Hedenmalm

Resum


The lattice of invariant subspaces of several Banach spaces of analytic functions on the unit disk, for example the Bergman spaces and the Dirichlet spaces, have been studied recently. A natural question is to what extent these investigations carry over to analogously defined spaces on an annulus. We consider this question in the context of general Banach spaces of analytic functions on finitely connected domains $\Omega$. The main result reads as follows: Assume that $B$ is a Banach space of analytic functions satisfying some conditions on the domain $\Omega$. Assume further that $M(B)$ is the set of all multipliers of $B$. Let $\Omega _1$ be a domain obtained from $\Omega$ by adding some of the bounded connectivity components of $\Bbb C \setminus \Omega$. Also, let $B_1$ be the closed subspace of $B$ of all functions that extend analytically to $\Omega _1$. Then the mapping $I \mapsto \operatorname{clos} (I\cdot M(B))$ gives a one-to-one correspondence between a class of multiplier invariant subspaces $I$ of $B_1$, and a class of multiplier invariant subspaces $J$ of $B$. The inverse mapping is given by $J \mapsto J \cap B_1$.

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