Negative Powers of Contractions Having a Strong AA+ Spectrum

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 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.

Keywords

• exceptional sets
• strong aa+-sets
• absolutely convergent fourier series
• absolutely convergent taylor series
• hyperfunctions
• primary 43a46, 46f15, 46j10
• secondary 42a20, 47a10