Using $H^2$ to find a cyclic vector in $\ell^2$

Question

Let us consider $\ell^p(\mathbb{Z})$. We know that the vector $e_1=(\dots,0,0,1,0,0,\dots)$ is a cyclic vector in sense that given the right shift operator $S:(\dots,x_0,x_1,x_2,\dots)\mapsto (\dots,x_{-1},x_0,x_1,\dots)$, $$\overline{\text{span}\{S^ne_1\}_{n\in\mathbb{Z}}}=\ell^2(\mathbb{Z}).$$

I wonder if the requirement for $n\in\mathbb{Z}$ is essential. I assume that we can find a vector $\vec{c}\in\ell^2(\mathbb{Z})$ such that

$$\overline{\text{span}\{S^n\vec c\}_{n\in\mathbb{N}}}=\ell^2(\mathbb{Z}).$$

I think that using $H^2(\mathbb{T})$ can be useful but I don't know how to use it. Am I right? Can you give an example of such vector?

Answer 1

It's possible to prove the existence of such a vector using Hardy space methods, though the relevant space is $H^1(\mathbb T)$, not $H^2(\mathbb T)$. Taking Fourier transforms, we are looking for an $f\in L^2(\mathbb T)$ such that the space $$ M=\text{span}\{z^nf(z):n\geq 0\}$$ is dense in $L^2(\mathbb T)$. I claim that any $f\in L^2(\mathbb T)$ such that 1) $f$ is nonzero almost everywhere, and 2) $\log|f|\notin L^1(\mathbb T)$, does the job. Trivially, we must have $f(z)\neq 0$ for Lebesgue a.e. $z\in\mathbb T$. From now on consider only these $f$. If the associated $M$ is not dense, then there exists a nonzero $g\in L^2(\mathbb T)$ such that $$ \int_\mathbb{T} z^n f(z)\overline{g(z)}\, dm(z)=0 $$ for all $n\geq 0$. This means that the function $h=\overline{f}g$ is nonzero and belongs to $H^1(\mathbb T)$, and is hence log integrable, i.e. $\int_\mathbb{T} \log|h(z)|\, dm(z) >-\infty$. Since $f, g$ belong to $L^2$, they also belong to $L^1$, and therefore $\int \log|f|, \int\log |g| <+\infty$. So we must have $\int_\mathbb{T}\log|f(z)|\, dm(z)>-\infty$ also. Thus if $\int \log|f|=-\infty$, we see that $M$ is dense.