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This article gives a description of invariant subspaces for the backwardshift generated by vector valued lacunary series and by a class of lacunarypower series in $H^2(\mathbb{D}, X)$, (where $X$ is an Hilbert space). Inparticular, we show that these series $f$ in $H^2(\mathbb{D}, X)$ are cyclicvectors 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 somefunctions whose spectrum is a finite union of lacunary sequences and in thepolydisc. In the scalar case $H^2$, we give a criterion on the Fourier spectrumof the function to have cyclicity for any power of the backward shift.

Lacunarity and cyclic vectors for the Backward Shift