Invariant subspaces for Bishop operators and beyond
Bishop operators $T_\alpha$ acting on $L^2[0,1)$ were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. We prove that all the Bishop operators are biquasitriangular and, derive as a consequence that they are norm limits...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/129026 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/129026 |
| Access Level: | acceso abierto |
| Palabra clave: | Bishop operators Invariant subspace problem Dunford’s property (C) Análisis funcional y teoría de operadores Teoría de números 1202.03 Álgebra y Espacios de Banach 1205.03 Problemas Diofánticos 1202.14 Espacio de Hilbert |
| Sumario: | Bishop operators $T_\alpha$ acting on $L^2[0,1)$ were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. We prove that all the Bishop operators are biquasitriangular and, derive as a consequence that they are norm limits of nilpotent operators. Moreover, by means of arithmetical techniques along with a theorem of Atzmon, the set of irrationals $\alpha \in (0,1)$ for which $T_\alpha$ is known to possess non-trivial closed invariant subspaces is considerably enlarged, extending previous results by Davie [11], MacDonald [21] and Flattot [14]. Furthermore, we essentially show that when our approach fails to produce invariant subspaces it is actually because Atzmon's Theorem cannot be applied. Finally, upon applying arithmetical bounds obtained, we deduce local spectral properties of Bishop operators proving, in particular, that neither of them satisfy Dunford's property (C). |
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