Abstract
The composition operator \(C_{\phi _a}f=f\circ \phi _a\) on the Hardy–Hilbert space \(H^2({\mathbb {D}})\) with affine symbol \(\phi _a(z)=az+1-a\) and \(0<a<1\) has the property that the Invariant Subspace Problem for complex separable Hilbert spaces holds if and only if every minimal invariant subspace for \(C_{\phi _a}\) is one-dimensional. These minimal invariant subspaces are always singly-generated \( K_f:= \overline{\textrm{span} \{f, C_{\phi _a}f, C^2_{\phi _a}f, \ldots \}}\) for some \(f\in H^2({\mathbb {D}})\). In this article we characterize the minimal \(K_f\) when f has a nonzero limit at the point 1 or if its derivative \(f'\) is bounded near 1. We also consider the role of the zero set of f in determining \(K_f\). Finally we prove a result linking universality in the sense of Rota with cyclicity.
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Acknowledgements
This work constitutes a part of the doctoral thesis of the second author, partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq Brasil, under the supervision of the third named author.
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Communicated by Aurelian Gheondea.
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Carmo, J.R., Eidt, B.H. & Noor, S.W. Minimal Invariant Subspaces for an Affine Composition Operator. Complex Anal. Oper. Theory 18, 59 (2024). https://doi.org/10.1007/s11785-024-01501-9
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DOI: https://doi.org/10.1007/s11785-024-01501-9