Lacunarity and cyclic vectors for the Backward Shift

TL;DR

This paper characterizes invariant subspaces and cyclic vectors for the backward shift in vector-valued Hardy spaces using lacunary series, providing new criteria for cyclicity based on spectral properties.

Contribution

It introduces new characterizations of invariant subspaces and cyclic vectors for the backward shift in vector-valued Hardy spaces using lacunary series and spectral conditions.

Findings

Lacunary series generate cyclic vectors iff their tail coefficients span the space.

Results extend to functions with spectra as finite unions of lacunary sequences.

Provides spectral criteria for cyclicity of functions under powers of the backward shift.

Abstract

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.