Cyclic vectors for backward hyponormal weighted shifts

Abstract:

A unilateral weighted shift $T$ on a Hilbert space $H$ is an operator such that $T{e_n} = {w_n}{e_{n + 1}}$ for some orthonormal basis $\left \{ {{e_n}} \right \}_{n = 0}^\infty$ and weight sequence $\left \{ {{w_n}} \right \}_{n = 0}^\infty$. If we assume ${w_n} > 0$, for all $n$, and let $\beta \left ( n \right ) = {w_0} \cdots {w_{n - 1}}$ for $n > 0$ and $\beta \left ( 0 \right ) = 1$, then $T$ is unitarily equivalent to $f \mapsto zf$ on the weighted space ${H^2}\left ( \beta \right )$ of formal power series $\sum \nolimits _{n = 0}^\infty {\hat f\left ( n \right ){z^n}}$ such that $\sum \nolimits _{n = 0}^\infty {{{\left | {\hat f\left ( n \right )} \right |}^2}{{\left [ {\beta \left ( n \right )} \right ]}^2} < \infty }$. Regarding $T$ as multiplication for $z$ on the space ${H^2}\left ( \beta \right )$, it is shown that, if ${w_n} \uparrow 1$ and $f$ is analytic in a neighborhood of the unit disk, then either $f$ is cyclic for ${T^ * }$ or $f$ is contained in a finite-dimensional ${T^ * }$-invariant subspace. This was shown—by different methods— for the unweighted shift operator by Douglas, Shields, and Shapiro [2]. It is also shown that every finite-dimensional ${T^ * }$-invariant subspace is of the form \[ {\left ( {{{\left ( {z - {\alpha _1}} \right )}^{{n_1}}} \cdots {{\left ( {z - {\alpha _k}} \right )}^{{n_k}}}{H^2}\left ( \beta \right )} \right )^ \bot },\] for some ${\alpha _1}, \ldots ,{\alpha _k}$ in the unit disk and ${n_1}, \ldots ,{n_k}$ positive integer.