Abstract. In this note we give several examples of invertible operators $T$ on Hilbert space such that the sets ${\cal C}(T)$ and ${\cal C}(T^{-1})$ of cyclic vectors for $T$ and $T^{-1}$, respectively, are different. This forecloses one possible approach to solving the famous problem of Halmos: if $T$ has a nontrivial invariant subspace, then does necessarily $T^{-1}$ have one too?

AMS Subject Classification (1991): 47A15

Keyword(s): Invariant subspaces

Received May 4, 2005, and in revised form September 20, 2005. (Registered under 5896/2009.)