# Invariant subspaces for Bishop operators and beyond

## TL;DR

This paper investigates Bishop operators, proving they are biquasitriangular and limits of nilpotent operators, and extends the set of irrationals for which these operators have non-trivial invariant subspaces, revealing new spectral properties.

## Contribution

It establishes the biquasitriangularity of Bishop operators, enlarges the class of irrationals with known invariant subspaces, and analyzes their spectral properties using arithmetical techniques.

## Key findings

- Bishop operators are biquasitriangular.
- They are norm limits of nilpotent operators.
- The set of irrationals with known invariant subspaces is expanded.

## Abstract

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, MacDonald and Flattot. Furthermore, we essentially show that when our approach fails to produce invariant subspaces it is actually because Atzmon 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 the Dunford property $(C)$.