The Method of the Weakly Conjugate Operator: Extensions and Applications to Operators on Graphs and Groups

Marius Mantoiu

Description

Niveau: Supérieur, Doctorat, Bac+8
The Method of the Weakly Conjugate Operator: Extensions and Applications to Operators on Graphs and Groups M. Ma˘ntoiu?, S. Richard?? and R. Tiedra de Aldecoa??? ? Departamento de Matematicas, Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile, email: ?? Dept. of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, United Kingdom, email: ??? CNRS (UMR 8088) and Dept. of Mathematics, University of Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France, email: Abstract In this review we present some recent extensions of the method of the weakly conjugate operator. We illustrate these developments through examples of operators on graphs and groups. Introduction In spectral analysis, one of the most powerful tools is the method of the conjugate operator, also called Mourre's commutator method after the seminal work of Mourre in the early eighties. This approach has reached a very high degree of precision and abstraction in [1]; see also [14] for further developments. In order to study the nature of the spectrum of a selfadjoint operator H , the main idea of the standard method is to find an auxiliary selfadjoint operator A such that the commutator i[H,A] is strictly positive when localized in some interval of the spectrum ofH .

• limiting absorption


• through examples


• operators acting


• absorption principle


• continuous


• natural continuous embeddings


• has ∑


• k?g ofh


• h0 has


• operators

Sujets

• Kingdom
• Jordan
• École centrale de Lyon
• Tiedra
• Synergy (logiciel)

Informations

• Publié par: mijec
• Publié le: 01 novembre 1918
• Nombre de lectures: 12
• Langue: English

Extrait

TheMethodoftheWeaklyConjugateOperator: ExtensionsandApplicationstoOperatorsonGraphsandGroups M.Ma˘ntoiu ∗ ,S.Richard ∗∗ andR.TiedradeAldecoa ∗∗∗ ∗ DepartamentodeMatema ´ ticas,UniversidaddeChile,LasPalmeras3425, Casilla653,Santiago,Chile,email:[e-mail] ∗∗ Dept.ofPureMathematicsandMathematicalStatistics,CentreforMathematical Sciences,UniversityofCambridge,Cambridge,CB30WB,UnitedKingdom, email:[e-mail] ∗∗∗ CNRS(UMR8088)andDept.ofMathematics,UniversityofCergy-Pontoise, 2avenueAdolpheChauvin,95302Cergy-PontoiseCedex,France, email:[e-mail] Abstract Inthisreviewwepresentsomerecentextensionsofthemethodoftheweaklyconjugateoperator. Weillustratethesedevelopmentsthroughexamplesofoperatorsongraphsandgroups. Introduction Inspectralanalysis,oneofthemostpowerfultoolsisthemethodoftheconjugateoperator,also calledMourre’scommutatormethodaftertheseminalworkofMourreintheearlyeighties.This approachhasreachedaveryhighdegreeofprecisionandabstractionin[1];seealso[14]for furtherdevelopments.InordertostudythenatureofthespectrumofaselfadjointoperatorH, themainideaofthestandardmethodistofindanauxiliaryselfadjointoperatorAsuchthatthe commutatori[H,A]isstrictlypositivewhenlocalizedinsomeintervalofthespectrumofH.More precisely,onelooksforintervalsJof R suchthat E(J)i[H,A]E(J) ≥ aE(J)(1) forsomestrictlypositiveconstant a thatdependson J ,where E(J) denotesthespectralprojection of H ontheinterval J .Anadditionalcompactcontributionto(1)isallowed,greatlyenlargingthe rangeofapplications. Whenstrictpositivityisnotavailable,onecaninsteadlookforanAsuchthatthecommutatoris positiveandinjective,i.e. i[H,A]>0.(2) ThisrequirementisclosetotheoneoftheKato-Putnamtheorem,cf.[29,Thm.XIII.28].A newcommutatormethodbasedonsuchaninequalitywasproposedin[7,8].Byanalogytothe ∗ OnleavefromUniversite´deLyon;Universite´Lyon1;INSAdeLyon,F-69621;EcoleCentraledeLyon;CNRS, UMR5208,InstitutCamilleJordan,43blvddu11novembre1918,F-69622Villeurbanne-Cedex,France. 1