  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 $(C)$.
