Abstract
A closed subspace is invariant under the Cesàro operator \({\cal C}\) on the classical Hardy space \({H^2}(\mathbb{D})\) if and only if its orthogonal complement is invariant under the C0-semigroup of composition operators induced by the affine maps \({\varphi _t}(z) = {e^{ - t}}z + 1 - {e^{ - t}}\) for t ≥ 0 and \(z =\mathbb{D}\). The corresponding result also holds in the Hardy spaces Hp(\(\mathbb{D}\)) for 1 < p < ∞. Moreover, in the Hilbert space setting, by linking the invariant subspaces of \({\cal C}\) to the lattice of the closed invariant subspaces of the standard right-shift semigroup acting on a particular weighted L2-space on the line, we exhibit a large class of non-trivial closed invariant subspaces and provide a complete characterization of the finite codimensional ones, establishing, in particular, the limits of such an approach towards describing the lattice of all invariant subspaces of \({\cal C}\). Finally, we present a functional calculus argument which allows us to extend a recent result by Mashreghi, Ptak and Ross regarding the square root of \({\cal C}\) and discuss its invariant subspaces.
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The authors are grateful to a referee for carefully reading the manuscript and providing some extremely helpful comments which improved its readability.
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Both authors are partially supported by Plan Nacional I+D grant no. PID2019-105979GB-I00, Spain. The first author is also supported by the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001).
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Gallardo-Gutiérrez, E.A., Partington, J.R. Insights on the Cesàro operator: shift semigroups and invariant subspaces. JAMA 152, 595–614 (2024). https://doi.org/10.1007/s11854-023-0305-0
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DOI: https://doi.org/10.1007/s11854-023-0305-0