The spectrum of some compressions of unilateral shifts

Let $E$ be a star-shaped Banach space of analytic functions on the open unit disc $\mathbb D$. We assume that the unilateral shift $S\colon z\to zf$ and the backward shift $T\colon f\to\frac{f-f(0)}{z}$ are bounded on $E$ and that their spectrum is the closed unit disc.
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 shows that if $g$ has an analytic extension to $\mathbb D\cup D(\zeta,r)$ for some $r>0$, with $g(\zeta)\ne 0$, 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 $S_M\colon f+M\to zf+M$ of the unilateral shift to the quotient space $E/M$. This shows in particular that $\operatorname{Spec}(S_M)=\{1\}$ for some $z$-invariant subspaces $M$ of weighted Hardy spaces constructed by N. K. Nikol'skiĭ in the seventies by using the Keldysh method.