<abstract xmlns="http://www.w3.org/1999/xhtml">

Zarrabi proved in 1993 that if the spectrum of a contraction <italic>T</italic> on a Banach space is a countable subset of the unit circle 𝕋, and if 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mo>lim</mo></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></msub><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mrow><mrow><mo>‖</mo> <mrow><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow> <mo>‖</mo></mrow></mrow><mo>)</mo></mrow></mrow><mrow><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math>
<tex-math>{\lim _{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {\sqrt n }} = 0</tex-math>
</alternatives>
</inline-formula>, then <italic>T</italic> is an isometry, so that ‖<italic>Tn</italic>‖ = 1 for every <italic>n</italic> ∈ ℤ. It is also known that if <italic>C</italic> is the usual triadic Cantor set then every contraction <italic>T</italic> on a Banach space such that <italic>Spec</italic>(<italic>T</italic> ) ⊂ 𝒞 satisfying 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>lim</mo><mi> </mi><mi>s</mi><mi>u</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></msub><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mrow><mrow><mo>‖</mo> <mrow><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow> <mo>‖</mo></mrow></mrow><mo>)</mo></mrow></mrow><mrow><msup><mrow><mi>n</mi></mrow><mi>α</mi></msup></mrow></mfrac><mo>&lt;</mo><mo>+</mo><mo>∞</mo></mrow></math>
<tex-math>\lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} &lt; + \infty</tex-math>
</alternatives>
</inline-formula>
 for some 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>&lt;</mo><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mo>−</mo><mo>log</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi> </mi><mo>log</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mo>−</mo><mo>log</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></mfrac></mrow></math>
<tex-math>\alpha &lt; {{\log \left( 3 \right) - \log \left( 2 \right)} \over {2\,\log \left( 3 \right) - \log \left( 2 \right)}}</tex-math>
</alternatives>
</inline-formula> is an isometry.
In the other direction an easy refinement of known results shows that if a closed <italic>E</italic> ⊂ 𝕋 is not a “strong <italic>AA</italic>+-set” then for every sequence (<italic>un)n</italic>≥1 of positive real numbers such that lim inf<italic>n</italic>→+∞<italic>un</italic> = + ∞ there exists a contraction <italic>T</italic> on some Banach space such that <italic>Spec</italic>(<italic>T</italic> )⊂ <italic>E</italic>, ‖<italic>T</italic>−<italic>n</italic>‖ = <italic>O</italic>(<italic>u</italic>n) as <italic>n</italic> → + ∞ and supn≥1 ‖<italic>T</italic>−n‖ = + ∞.
We show conversely that if <italic>E</italic> ⊂ 𝕋 is a strong <italic>AA</italic>+-set then there exists a nondecreasing unbounded sequence (<italic>u</italic>n)n≥1 such that for every contraction <italic>T</italic> on a Banach space satsfying <italic>Spec</italic>(<italic>T</italic>) ⊂ <italic>E</italic> and ‖<italic>T</italic>−n ‖ = <italic>O</italic>(<italic>u</italic>n) as <italic>n</italic> → + ∞ we have <italic>supn</italic>>0 ‖<italic>T</italic>−n ‖ ≤ <italic>K</italic>, where <italic>K &lt;</italic> + ∞ denotes the “<italic>AA</italic>+-constant” of <italic>E</italic> (closed countanble subsets of 𝕋 and the triadic Cantor set are strong <italic>AA</italic>+-sets of constant 1).
</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.
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).

Negative Powers of Contractions Having a Strong <italic>AA</italic>+ Spectrum

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Negative Powers of Contractions Having a Strong  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 ‖)nα0 ‖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).

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...

Negative Powers of Contractions Having a Strong AA⁺ Spectrum

Published Online: Jun 07, 2023

Page range: 209 - 215

Received: Sep 20, 2022

Accepted: Nov 22, 2022

DOI: https://doi.org/10.2478/mjpaa-2023-0015

Keywords
exceptional sets, strong -sets, absolutely convergent Fourier series, absolutely convergent Taylor series, hyperfunctions

© 2023 Jean Esterle, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Negative Powers of Contractions Having a Strong AA+ Spectrum

Published Online: Jun 07, 2023

Page range: 209 - 215

Received: Sep 20, 2022

Accepted: Nov 22, 2022

DOI: https://doi.org/10.2478/mjpaa-2023-0015

Keywordsexceptional sets, strong -sets, absolutely convergent Fourier series, absolutely convergent Taylor series, hyperfunctions

© 2023 Jean Esterle, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Negative Powers of Contractions Having a Strong AA⁺ Spectrum

Keywords
exceptional sets, strong -sets, absolutely convergent Fourier series, absolutely convergent Taylor series, hyperfunctions