Lacunarity and cyclic vectors for the Backward Shift

This article gives a description of invariant subspaces for the backward shift generated by vector valued lacunary series and by a class of lacunary power series in $H^2(\mathbb{D}, X)$, (where $X$ is an Hilbert space). In particular, we show that these series $f$ in $H^2(\mathbb{D}, X)$ are cyclic vectors if and only if the queue of Taylor coefficients $\{\hat{f}(k)$, $k>N\}$ generates the whole space $X$. Analogues of this result are obtained for some functions whose spectrum is a finite union of lacunary sequences and in the polydisc. In the scalar case $H^2$, we give a criterion on the Fourier spectrum of the function to have cyclicity for any power of the backward shift.