A new proof of Poltoratskii's theorem

Vojkan Jakšić; Yoram Last

doi:10.1016/j.jfa.2003.09.014

A new proof of Poltoratskii's theorem

Vojkan Jakšić^*, Yoram Last

^*Corresponding author for this work

Einstein Institute of Mathematics

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

19 Scopus citations

Abstract

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on ℝ and any f∈L¹(ℝ,dμ),lim_ε→0 (F_fu(E+iε)/F_μ(E+iε))=f(E) a.e. w.r.t. μ_sing, where μ_sing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, F_fμ(z)=∫(x-z)^-1 f(x) dμ(x) and F_μ(z)=∫(x-z)^-1dμ(x).

Original language	English
Pages (from-to)	103-110
Number of pages	8
Journal	Journal of Functional Analysis
Volume	215
Issue number	1
DOIs	https://doi.org/10.1016/j.jfa.2003.09.014
State	Published - 1 Oct 2004

Bibliographical note

Funding Information:
V.J.’s work was partially supported by NSERC. Y.L.’s work was partially supported by The Israel Science Foundation (Grant 188/02). We thank Barry Simon for motivating this work and for useful discussions.

Keywords

Borel transform
Poltoratskii theorem
Rank one perturbations
Singular measure
Stieltjes transform

Access to Document

10.1016/j.jfa.2003.09.014

Cite this

@article{d5817d7ae74f41abac1dd85e4f6f7818,

title = "A new proof of Poltoratskii's theorem",

abstract = "We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on ℝ and any f∈L1(ℝ,dμ),limε→0 (Ffu(E+iε)/Fμ(E+iε))=f(E) a.e. w.r.t. μsing, where μsing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, Ffμ(z)=∫(x-z)-1 f(x) dμ(x) and Fμ(z)=∫(x-z)-1dμ(x).",

keywords = "Borel transform, Poltoratskii theorem, Rank one perturbations, Singular measure, Stieltjes transform",

author = "Vojkan Jak{\v s}i{\'c} and Yoram Last",

note = "Funding Information: V.J.{\textquoteright}s work was partially supported by NSERC. Y.L.{\textquoteright}s work was partially supported by The Israel Science Foundation (Grant 188/02). We thank Barry Simon for motivating this work and for useful discussions.",

year = "2004",

month = oct,

day = "1",

doi = "10.1016/j.jfa.2003.09.014",

language = "אנגלית",

volume = "215",

pages = "103--110",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - A new proof of Poltoratskii's theorem

AU - Jakšić, Vojkan

AU - Last, Yoram

N1 - Funding Information: V.J.’s work was partially supported by NSERC. Y.L.’s work was partially supported by The Israel Science Foundation (Grant 188/02). We thank Barry Simon for motivating this work and for useful discussions.

PY - 2004/10/1

Y1 - 2004/10/1

N2 - We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on ℝ and any f∈L1(ℝ,dμ),limε→0 (Ffu(E+iε)/Fμ(E+iε))=f(E) a.e. w.r.t. μsing, where μsing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, Ffμ(z)=∫(x-z)-1 f(x) dμ(x) and Fμ(z)=∫(x-z)-1dμ(x).

AB - We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on ℝ and any f∈L1(ℝ,dμ),limε→0 (Ffu(E+iε)/Fμ(E+iε))=f(E) a.e. w.r.t. μsing, where μsing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, Ffμ(z)=∫(x-z)-1 f(x) dμ(x) and Fμ(z)=∫(x-z)-1dμ(x).

KW - Borel transform

KW - Poltoratskii theorem

KW - Rank one perturbations

KW - Singular measure

KW - Stieltjes transform

UR - http://www.scopus.com/inward/record.url?scp=4344689487&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2003.09.014

DO - 10.1016/j.jfa.2003.09.014

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

AN - SCOPUS:4344689487

SN - 0022-1236

VL - 215

SP - 103

EP - 110

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

ER -

A new proof of Poltoratskii's theorem

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this