Norm estimates of almost Mathieu operators

Florin P. Boca; Alexandru Zaharescu

doi:10.1016/j.jfa.2004.09.013

Norm estimates of almost Mathieu operators

Florin P. Boca
, Alexandru Zaharescu

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We estimate the norm of the almost Mathieu operator H_θ,λ = U_θ + U_θ* + (λ/2)(V_θ + V_θ*), regarded as an element in the rotation C*-algebra A_θ=C*(U_θ, V_θ unitaries : U_θ V_θ = e^2πiθV_θU_θ). In the process, we prove for every λ ∈ ℝ and θ ∈ [1/4, 1/2] the inequality A formula is presented. This significantly improves the inequality ∥H_θ,2∥ ≤ 2 2, θ ∈ [1/4, 1/2], conjectured by Béguin, Valette and Zuk.

Original language	English (US)
Pages (from-to)	76-96
Number of pages	21
Journal	Journal of Functional Analysis
Volume	220
Issue number	1
DOIs	https://doi.org/10.1016/j.jfa.2004.09.013
State	Published - Mar 1 2005

Keywords

Almost Mathieu operators
Norms
Rotation C*-algebras

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2004.09.013

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title = "Norm estimates of almost Mathieu operators",

abstract = "We estimate the norm of the almost Mathieu operator Hθ,λ = Uθ + Uθ* + (λ/2)(Vθ + Vθ*), regarded as an element in the rotation C*-algebra Aθ=C*(Uθ, Vθ unitaries : Uθ Vθ = e2πiθVθUθ). In the process, we prove for every λ ∈ ℝ and θ ∈ [1/4, 1/2] the inequality A formula is presented. This significantly improves the inequality ∥Hθ,2∥ ≤ 2 2, θ ∈ [1/4, 1/2], conjectured by B{\'e}guin, Valette and Zuk.",

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author = "Boca, \{Florin P.\} and Alexandru Zaharescu",

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doi = "10.1016/j.jfa.2004.09.013",

language = "English (US)",

volume = "220",

pages = "76--96",

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TY - JOUR

T1 - Norm estimates of almost Mathieu operators

AU - Boca, Florin P.

AU - Zaharescu, Alexandru

PY - 2005/3/1

Y1 - 2005/3/1

N2 - We estimate the norm of the almost Mathieu operator Hθ,λ = Uθ + Uθ* + (λ/2)(Vθ + Vθ*), regarded as an element in the rotation C*-algebra Aθ=C*(Uθ, Vθ unitaries : Uθ Vθ = e2πiθVθUθ). In the process, we prove for every λ ∈ ℝ and θ ∈ [1/4, 1/2] the inequality A formula is presented. This significantly improves the inequality ∥Hθ,2∥ ≤ 2 2, θ ∈ [1/4, 1/2], conjectured by Béguin, Valette and Zuk.

AB - We estimate the norm of the almost Mathieu operator Hθ,λ = Uθ + Uθ* + (λ/2)(Vθ + Vθ*), regarded as an element in the rotation C*-algebra Aθ=C*(Uθ, Vθ unitaries : Uθ Vθ = e2πiθVθUθ). In the process, we prove for every λ ∈ ℝ and θ ∈ [1/4, 1/2] the inequality A formula is presented. This significantly improves the inequality ∥Hθ,2∥ ≤ 2 2, θ ∈ [1/4, 1/2], conjectured by Béguin, Valette and Zuk.

KW - Almost Mathieu operators

KW - Norms

KW - Rotation C-algebras

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UR - https://www.scopus.com/pages/publications/12444266896#tab=citedBy

U2 - 10.1016/j.jfa.2004.09.013

DO - 10.1016/j.jfa.2004.09.013

M3 - Article

AN - SCOPUS:12444266896

SN - 0022-1236

VL - 220

SP - 76

EP - 96

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

ER -

Norm estimates of almost Mathieu operators

Abstract

Keywords

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