Negative Powers of Contractions Having a Strong AA+ Spectrum

Abstract

Abstract 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 α0 ‖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).