Insights on the Cesàro operator: shift semigroups and invariant subspaces
A closed subspace is invariant under the Cesàro operator C on the classical Hardy space H2 (D) if and only if its orthogonal complement is invariant under the C0-semigroup of composition operators induced by the affine maps φt(z) = e−t z + 1 − e −t for t ≥ 0 and z ∈ D. The corresponding result also...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/71717 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/71717 |
| Access Level: | acceso abierto |
| Palabra clave: | 517.98 Cesàro operator Composition operator Shift semigroup Invariant subspaces Functional calculus Análisis matemático 1202 Análisis y Análisis Funcional |
| Sumario: | A closed subspace is invariant under the Cesàro operator C on the classical Hardy space H2 (D) if and only if its orthogonal complement is invariant under the C0-semigroup of composition operators induced by the affine maps φt(z) = e−t z + 1 − e −t for t ≥ 0 and z ∈ D. The corresponding result also holds in the Hardy spaces Hp(D) for 1 < p < ∞. Moreover, in the Hilbert space setting, by linking the invariant subspaces of C to the lattice of the closed invariant subspaces of the standard right-shift semigroup acting on a particular weighted L 2 -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 C. Finally, we present a functional calculus which allows us to extend a recent result by Mashreghi, Ptak and Ross regarding the square root of C and discuss its invariant subspaces. |
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