Invariant subspaces for commuting contractions

It is Shown that each finite commuting system of contractions which possesses a unitary dilation and for which the Harte spectrum is dominating in the open unit polydisc possesses non-trivial joint invariant subspaces. Since by a well-known result of Ando each commuting pair of contractions admits a unitary dilation, we obtain in particular that each commuting pair of contractions with dominating Harte spectrum in the bidisc possesses non-trivial joint invariant subspaces.