On powers of operators with spectrum in cantor sets and spectral synthesis

2017/06/09 by Mohamed Zarrabi, Zarrabi, Mohamed
Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.1706.02943

openalex publication_date 2017/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

For ξ∈ ( 0, (1)/(2) ), let E_ξ be the perfect symmetric set associated with ξ, that is E_ξ = \ exp ( 2i π(1-ξ) ∑_{n = 1}^+∞ ε_n ξ^n-1 ) : ε_n = 0 \textrm or 1 (n ≥ 1) \ and b(ξ) = \fraclog\frac1ξ - log22log\frac1ξ - log2. Let q≥ 3 be an integer and s be a nonnegative real number. We show that any invertible operator T on a Banach space with spectrum contained in E_1/q that satisfies amp; amp; ‖ Tⁿ ‖ = O ( n^s ), n → +∞
amp; \textrmand amp; ‖ T^-n ‖ = O ( e^nβ ), n → +∞ \textrm for some βlt; b(1/q), also satisfies the stronger property ‖ T^-n ‖ = O ( n^s ), n → +∞. We also show that this result is false for E_ξ when 1/ξ is not a Pisot number and that the constant b(1/q) is sharp. As a consequence we prove that, if ω is a submulticative weight such that ω(n)=(1+n)^s, (n ≥ 0) and C^-1 (1+|n|)^s ≤ ω(-n) ≤ C e^nβ, (n≥ 0), for some constants C>0 and β< b( 1/q), then E_1/q satisfies spectral synthesis in the Beurling algebra of all continuous functions f on the unit circle \mathbbT such that ∑_{n = -∞}^+∞ | \widehatf(n) | ω(n) < +∞.

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