Chamizo, FernandoGallardo Gutiérrez, Eva AntoniaMonsalve López, MiguelUbis, Adrián2025-12-152025-12-152020Chamizo , F., Gallardo-Gutiérrez, E. A., Monsalve-López, M., Ubis, A. Invariant subspaces for Bishop operators and beyond. Adv. Math. 375, 107365 (2020)10.1016/j.aim.2020.107365https://hdl.handle.net/20.500.14352/129026Bishop 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).engAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/journal articlehttps://doi.org/10.1016/j.aim.2020.107365open accessBishop operatorsInvariant subspace problemDunford’s property (C)Análisis funcional y teoría de operadoresTeoría de números1202.03 Álgebra y Espacios de Banach1205.03 Problemas Diofánticos1202.14 Espacio de Hilbert