Zarrabi proved in 1993 that if the spectrum of a contraction T on a Banach space is a countable subset of the unit circle 𝕋, and if
{\lim _{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {\sqrt n }} = 0
, then T is an isometry, so that ‖Tn‖ = 1 for every n ∈ ℤ. It is also known that if C is the usual triadic Cantor set then every contraction T on a Banach space such that Spec(T ) ⊂ 𝒞 satisfying
\lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} < + \infty
for some
\alpha < {{\log \left( 3 \right) - \log \left( 2 \right)} \over {2\,\log \left( 3 \right) - \log \left( 2 \right)}}
is an isometry.
In the other direction an easy refinement of known results shows that if a closed E ⊂ 𝕋 is not a “strong AA+-set” then for every sequence (un)n≥1 of positive real numbers such that lim infn→+∞un = + ∞ there exists a contraction T on some Banach space such that Spec(T )⊂ E, ‖T−n‖ = O(un) as n → + ∞ and supn≥1 ‖T−n‖ = + ∞.
We show conversely that if E ⊂ 𝕋 is a strong AA+-set then there exists a nondecreasing unbounded sequence (un)n≥1 such that for every contraction T on a Banach space satsfying Spec(T) ⊂ E and ‖T−n ‖ = O(un) as n → + ∞ we have supn>0 ‖T−n ‖ ≤ K, where K < + ∞ denotes the “AA+-constant” of E (closed countanble subsets of 𝕋 and the triadic Cantor set are strong AA+-sets of constant 1).
