The spectrum of some compressions of unilateral shifts

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''

$$\sum_{n\ge 0}\frac{\log\Vert S^n\Vert+\log \Vert T^n\Vert}{ 1+n^2}<+\infty,$$

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.