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 lim n → + ∞ 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   s u p n → + ∞ log ( ‖ T − n ‖ ) n α < + ∞ \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 (un)n ≥1 of positive real numbers such that lim inf n →+∞ un = + ∞ there exists a contraction T on some Banach space such that Spec(T )⊂ E, ‖T −n ‖ = O(u n) 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 (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 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).