The spectrum of some compressions of unilateral shifts

Abstract:

Let $E$ be a star-shaped Banach space of analytic functions on the open unit disk $\mathbb D$. It is assumed that the unilateral shift $S : z\to zf$ and the backward shift $T : f\to \frac {f-f(0)}{ z}$ are bounded on $E$ and that their spectrum is the closed unit disk.

Let $M$ be a closed $z$-invariant subspace of $E$ such that $\dim (M/zM)=1$, and let $g\in M$. The main result of the paper states that if $g$ has an analytic extension to $\mathbb D \cup D(\zeta ,r)$ for some $r>0$, with $g(\zeta ) \neq 0$, and if $S$ and $T$ satisfy the “nonquasianalytic condition” \[ \sum _{n\ge 0}\frac {\log \| S^n\|+\log \| T^n\|}{ 1+n^2}<+\infty , \] then $\zeta$ does not belong to the spectrum of the compression $S_M : f+M\to zf +M$ of the unilateral shift to the quotient space $E/M$. This shows in particular that $\mathrm {Spec}(S_M)=\{1\}$ for some $z$-invariant subspaces $M$ of weighted Hardy spaces that were constructed by N. K. Nikol′skiĭ in the 1970s by using the Keldysh method.