RT Journal Article
T1 Invariant subspaces for Bishop operators and beyond
A1 Chamizo, Fernando
A1 Gallardo Gutiérrez, Eva Antonia
A1 Monsalve López, Miguel
A1 Ubis, Adrián
AB 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).
PB Elsevier
YR 2020
FD 2020
LK https://hdl.handle.net/20.500.14352/129026
UL https://hdl.handle.net/20.500.14352/129026
LA eng
NO Chamizo , F., Gallardo-Gutiérrez, E. A., Monsalve-López, M., Ubis, A. Invariant subspaces for Bishop operators and beyond. Adv. Math. 375, 107365 (2020)
NO Ministerio de Ciencia e Innovación
NO Universidad Complutense de Madrid
DS Docta Complutense
RD 22 ene 2026