Negative Powers of Contractions Having a Strong AA⁺ Spectrum

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 limn→+∞log(‖ T−n ‖)n=0 {\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 supn→+∞log(‖ T−n ‖)nα<+∞ \lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} < + \infty for some α<log(3)−log(2)2 log(3)−log(2) \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 (u_n)_n≥1 of positive real numbers such that lim inf_n→+∞u_n = + ∞ there exists a contraction T on some Banach space such that Spec(T )⊂ E, ‖T⁻ⁿ‖ = O(u_n) as n → + ∞ and sup_n≥1 ‖T⁻ⁿ‖ = + ∞.

We show conversely that if E ⊂ 𝕋 is a strong AA⁺-set then there exists a nondecreasing unbounded sequence (u_n)_n≥1 such that for every contraction T on a Banach space satsfying Spec(T) ⊂ E and ‖T⁻ⁿ ‖ = O(u_n) as n → + ∞ we have sup_n>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).