Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Uri Kaluzhny; Yoram Last

doi:10.1016/j.jfa.2010.05.014

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Uri Kaluzhny^*, Yoram Last

^*Corresponding author for this work

Einstein Institute of Mathematics

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

1 Scopus citations

Abstract

We consider random self-adjoint Jacobi matrices of the form. on ℓ²(N), where {an(ω)>0} and {bn(ω)∈R} are sequences of random variables on a probability space (ω,dP(ω)) such that there exists q∈N, such that for any l∈N,. are independent random variables of zero mean satisfying. Let J_p be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in J_ω. We prove that for a.e. ω, the a.c. spectrum of the operator J_ω equals to and fills the spectrum of J_p. If, moreover,. then for a.e. ω, the spectrum of J_ω is purely absolutely continuous on the interior of the bands that make up the spectrum of Jp.

Original language	English
Pages (from-to)	1029-1044
Number of pages	16
Journal	Journal of Functional Analysis
Volume	260
Issue number	4
DOIs	https://doi.org/10.1016/j.jfa.2010.05.014
State	Published - 28 Feb 2011

Bibliographical note

Funding Information:
We would like to thank J. Breuer, M. Shamis, and B. Simon for useful discussions. This research was supported in part by The Israel Science Foundation (Grant No. 1169/06) and by Grant 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel.

Keywords

Absolutely continuous spectrum
Random Jacobi matrices

Access to Document

10.1016/j.jfa.2010.05.014

Cite this

@article{4ec2db3dd42a471db3a2781dbb2c8e0a,

title = "Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation",

abstract = "We consider random self-adjoint Jacobi matrices of the form. on ℓ2(N), where {an(ω)>0} and {bn(ω)∈R} are sequences of random variables on a probability space (ω,dP(ω)) such that there exists q∈N, such that for any l∈N,. are independent random variables of zero mean satisfying. Let Jp be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in Jω. We prove that for a.e. ω, the a.c. spectrum of the operator Jω equals to and fills the spectrum of Jp. If, moreover,. then for a.e. ω, the spectrum of Jω is purely absolutely continuous on the interior of the bands that make up the spectrum of Jp.",

keywords = "Absolutely continuous spectrum, Random Jacobi matrices",

author = "Uri Kaluzhny and Yoram Last",

note = "Funding Information: We would like to thank J. Breuer, M. Shamis, and B. Simon for useful discussions. This research was supported in part by The Israel Science Foundation (Grant No. 1169/06) and by Grant 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel.",

year = "2011",

month = feb,

day = "28",

doi = "10.1016/j.jfa.2010.05.014",

language = "אנגלית",

volume = "260",

pages = "1029--1044",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "4",

}

TY - JOUR

T1 - Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

AU - Kaluzhny, Uri

AU - Last, Yoram

N1 - Funding Information: We would like to thank J. Breuer, M. Shamis, and B. Simon for useful discussions. This research was supported in part by The Israel Science Foundation (Grant No. 1169/06) and by Grant 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel.

PY - 2011/2/28

Y1 - 2011/2/28

N2 - We consider random self-adjoint Jacobi matrices of the form. on ℓ2(N), where {an(ω)>0} and {bn(ω)∈R} are sequences of random variables on a probability space (ω,dP(ω)) such that there exists q∈N, such that for any l∈N,. are independent random variables of zero mean satisfying. Let Jp be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in Jω. We prove that for a.e. ω, the a.c. spectrum of the operator Jω equals to and fills the spectrum of Jp. If, moreover,. then for a.e. ω, the spectrum of Jω is purely absolutely continuous on the interior of the bands that make up the spectrum of Jp.

AB - We consider random self-adjoint Jacobi matrices of the form. on ℓ2(N), where {an(ω)>0} and {bn(ω)∈R} are sequences of random variables on a probability space (ω,dP(ω)) such that there exists q∈N, such that for any l∈N,. are independent random variables of zero mean satisfying. Let Jp be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in Jω. We prove that for a.e. ω, the a.c. spectrum of the operator Jω equals to and fills the spectrum of Jp. If, moreover,. then for a.e. ω, the spectrum of Jω is purely absolutely continuous on the interior of the bands that make up the spectrum of Jp.

KW - Absolutely continuous spectrum

KW - Random Jacobi matrices

UR - http://www.scopus.com/inward/record.url?scp=78650305791&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2010.05.014

DO - 10.1016/j.jfa.2010.05.014

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:78650305791

SN - 0022-1236

VL - 260

SP - 1029

EP - 1044

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 4

ER -

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this