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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Invariant subspaces in Banach spaces of analytic functions


Author: Stefan Richter
Journal: Trans. Amer. Math. Soc. 304 (1987), 585-616
MSC: Primary 47B38; Secondary 46E15, 46J15
DOI: https://doi.org/10.1090/S0002-9947-1987-0911086-8
MathSciNet review: 911086
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Abstract: We study the invariant subspace structure of the operator of multiplication by $ z$, $ {M_z}$, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class $ {\mathcal{B}_1}(\overline \Omega )$. We say that an invariant subspace $ \mathcal{M}$ satisfies $ \operatorname{cod} \mathcal{M} = 1$ if $ z\mathcal{M}$ has codimension one in $ \mathcal{M}$. We give various conditions on invariant subspaces which imply that $ \operatorname{cod} \mathcal{M} = 1$. In particular, we give a necessary and sufficient condition on two invariant subspaces $ \mathcal{M}$, $ \mathcal{N}$ with $ \operatorname{cod} \mathcal{M} = \operatorname{cod} \mathcal{N} = 1$ so that their span again satisfies $ \operatorname{cod} (\mathcal{M} \vee \mathcal{N}) = 1$. This result will be used to show that any invariant subspace of the Bergman space $ L_a^p,\,p \geqslant 1$, which is generated by functions in $ L_a^{2p}$, must satisfy $ \operatorname{cod} \mathcal{M} = 1$. For an invariant subspace $ \mathcal{M}$ we then consider the operator $ S = M_z^{\ast}\vert{\mathcal{M}^ \bot }$. Under some extra assumption on the domain of holomorphy we show that the spectrum of $ S$ coincides with the approximate point spectrum iff $ \operatorname{cod} \mathcal{M} = 1$. Finally, in the last section we obtain a structure theorem for invariant subspaces with $ \operatorname{cod} \mathcal{M} = 1$. This theorem applies to Dirichlet-type spaces.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0911086-8
Article copyright: © Copyright 1987 American Mathematical Society