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Invariant subspaces on multiply connected domains. (English) Zbl 0927.47002
Summary: 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 \(\mathbb{C}\setminus\Omega\). Also, let \(B_1\) be the closed subspace of \(B\) of all functions that extend analytically to \(\Omega_1\). Then the mapping \(I\mapsto \text{clos}(I\cdot M(B))\) 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 \(J\mapsto J\cap B_1\).

MSC:
47A15 Invariant subspaces of linear operators
46H30 Functional calculus in topological algebras
46E15 Banach spaces of continuous, differentiable or analytic functions
47B38 Linear operators on function spaces (general)
30H05 Spaces of bounded analytic functions of one complex variable
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