Abstract. We present a self-contained method initiated by A. M. Davie to prove the existence of nontrivial hyperinvariant subspace for Bishop-type operator $T_\alpha $ on $L^2(0,1)$ associated with an irrational $\alpha\in (0,1)$. Using all the strength of the Denjoy--Carleman theorem, we prove that our method works except on a set of Hausdorff measure equal to zero. We also show how to construct Liouville numbers $\alpha $ for which $T_\alpha $ has nontrivial hyperinvariant subspaces.

AMS Subject Classification (1991): 47A15; 47A10, 47A60

Received May 9, 2007, and in final form November 12, 2007. (Registered under 6040/2009.)