Summary: M. Zarrabi [Ann. Inst. Fourier 43, No. 1, 251–263 (1993; Zbl 0766.47002)] proved in 1993 that if the spectrum of a contraction \(T\) on a Banach space is a countable subset of the unit circle \(\mathbb{T}\), and if \(\lim_{n \to + \infty} \frac{\log (\|T^{-n}\|)}{\sqrt n} = 0\), then \(T\) is an isometry, so that \(\|T^n\| = 1\) for every \(n \in \mathbb{Z}\). It is also known that if \(\mathcal{C}\) is the usual triadic Cantor set then every contraction \(T\) on a Banach space such that \(\mathrm{Spec} (T) \subset \mathcal{C}\) satisfying \(\limsup_{n \to + \infty} \frac{\log (\|T^{-n}\|)}{n^\alpha} < +\infty\) for some \(\alpha < \frac{\log (3) - \log (2)}{2 \log (3) - \log (2)}\) is an isometry.
In the other direction an easy refinement of known results shows that if a closed \(E \subset \mathbb{T}\) is not a “strong \(AA^+\)-set” then for every sequence \((u_n)_{n \geq 1}\) of positive real numbers such that \(\liminf_{n \to +\infty} u_n = +\infty\) there exists a contraction \(T\) on some Banach space such that \(\mathrm{Spec} (T) \subset E\), \(\|T^{- n}\| = O(u_n)\) as \(n \to + \infty\) and \(\sup_{n \geq 1} \|T^{- n}\| = +\infty\).
We show conversely that if \(E \subset \mathbb{T}\) is a strong \(AA^+\)-set then there exists a nondecreasing unbounded sequence \((u_n)_{n \geq 1}\) such that for every contraction \(T\) on a Banach space satisfying \(\mathrm{Spec} (T) \subset E\) and \(\|T^{- n}\| = O(u_n)\) as \(n \to + \infty\) we have \(\sup_{n>0} \|T^{- n}\| \leq K\), where \(K < +\infty\) denotes the “\(AA^+\)-constant” of \(E\) (closed countanble subsets of \(\mathbb{T}\) and the triadic Cantor set are strong \(AA^+\)-sets of constant 1).