DilationofcommutingoperatorsBKDasIndianInstituteofTechnologyBombayNo - PDF document

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DilationofcommutingoperatorsBKDasIndianInstituteofTechnologyBombayNo - PPT Presentation

BKDas Dilationofcommutingoperators Introduction AnoperatorTonaHilbertspaceHisnormalifTT3T3T Normaloperatorsarewellunderstoodusingspectraltheory QuestionHowtostudyoperatorswhicharenotnorma ID: 949691

dilationofcommutingoperators das vonneumanninequality theorem das dilationofcommutingoperators theorem vonneumanninequality foranypolynomialp2c withrespecttothedecompositionk nition x0000 question howtostudyoperatorswhicharenotnormal understoodusingspectraltheory normaloperatorsarewell lettbeacontractiononh aunitaryuonk anoperatortonahilbertspacehisnormaliftt

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DilationofcommutingoperatorsB.K.DasIndianInstituteofTechnologyBombayNovember2 B.K.Das Dilationofcommutingoperators Introduction AnoperatorTonaHilbertspaceHisnormalifTT=TT. Normaloperatorsarewell-understoodusingspectraltheory. Question:Howtostudyoperatorswhicharenotnormal? AnoperatorUonaHilbertspaceHisaunitaryifUU=UU=IH. Denition LetTbeacontractiononH.Aunitar

yUonKHisadilationofTif(?)T=PHUjH,i.e.U=TwithrespecttothedecompositionK=HH?. B.K.Das Dilationofcommutingoperators Introduction AnoperatorTonaHilbertspaceHisnormalifTT=TT. Normaloperatorsarewell-understoodusingspectraltheory. Question:Howtostudyoperatorswhicharenotnormal? AnoperatorUonaHilbertspaceHisaunitaryifUU=UU=IH. Denition

LetTbeacontractiononH.AunitaryUonKHisadilationofTif(?)T=PHUjH,i.e.U=TwithrespecttothedecompositionK=HH?. B.K.Das Dilationofcommutingoperators Introduction AnoperatorTonaHilbertspaceHisnormalifTT=TT. Normaloperatorsarewell-understoodusingspectraltheory. Question:Howtostudyoperatorswhicharenotnormal? AnoperatorUonaHilbertspaceHisaunitaryifUU&

#3;=UU=IH. Denition LetTbeacontractiononH.AunitaryUonKHisadilationofTif(?)T=PHUjH,i.e.U=TwithrespecttothedecompositionK=HH?. B.K.Das Dilationofcommutingoperators Introduction AnoperatorTonaHilbertspaceHisnormalifTT=TT. Normaloperatorsarewell-understoodusingspectraltheory. Question:Howtostudyoperatorswhicharenotnormal? AnoperatorUon

aHilbertspaceHisaunitaryifUU=UU=IH. Denition LetTbeacontractiononH.AunitaryUonKHisadilationofTif(?)T=PHUjH,i.e.U=TwithrespecttothedecompositionK=HH?. B.K.Das Dilationofcommutingoperators Introduction AnoperatorTonaHilbertspaceHisnormalifTT=TT. Normaloperatorsarewell-understoodusingspectraltheory. Question:Howtostudyoperatorswhi

charenotnormal? AnoperatorUonaHilbertspaceHisaunitaryifUU=UU=IH. Denition LetTbeacontractiononH.AunitaryUonKHisadilationofTif(?)T=PHUjH,i.e.U=TwithrespecttothedecompositionK=HH?. B.K.Das Dilationofcommutingoperators Introduction AnoperatorTonaHilbertspaceHisnormalifTT=TT. Normaloperatorsarewell-understoodusingspectraltheory. Qu

estion:Howtostudyoperatorswhicharenotnormal? AnoperatorUonaHilbertspaceHisaunitaryifUU=UU=IH. Denition LetTbeacontractiononH.AunitaryUonKHisadilationofTifTn=PHUnjHforalln2N,i.e.Un=TnwithrespecttothedecompositionK=HH?.Inthiscase,p(T)=PHp(U)jHforanypolynomialp2C[z]. B.K.Das Dilationofcommutingoperators Denition LetT=(T1;:::;Td)bead-

tupleofcommutingcontractionsonH.Ad-tupleofcommutingunitaryU=(U1;:::;Ud)onKHisadilationofTifp(T)=PHp(U)jHforanypolynomialp2C[z1;:::;zd],i.e.p(U)=p(T)withrespecttothedecompositionK=HH?. B.K.Das Dilationofcommutingoperators Theorem(Nagy-Foias) LetTbeacontractiononaHilbertspaceH.ThenThasauniqueminimalunitarydilation. vonNeumanninequality:Foranypol

ynomialp2C[z],kp(T)ksupz2Djp(z)j: Theorem(T.Ando) Let(T1;T2)beapairofcommutingcontractionsonH.Then(T1;T2)dilatestoapairofcommutingunitaries(U1;U2). vonNeumanninequality:Foranypolynomialp2C[z1;z2],kp(T1;T2)ksup(z1;z2)2D2jp(z1;z2)j: NeitherdilationnorthevonNeumanninequalityholdsford-tuplesofcommutingcontractionswithd�2. B.K.Das Dilationofcommutingoperato

rs Theorem(Nagy-Foias) LetTbeacontractiononaHilbertspaceH.ThenThasauniqueminimalunitarydilation. vonNeumanninequality:Foranypolynomialp2C[z],kp(T)ksupz2Djp(z)j: Theorem(T.Ando) Let(T1;T2)beapairofcommutingcontractionsonH.Then(T1;T2)dilatestoapairofcommutingunitaries(U1;U2). vonNeumanninequality:Foranypolynomialp2C[z1;z2],kp(T1;T2)ksup(z1;z2)2D2jp(z1;z2)j:

5;NeitherdilationnorthevonNeumanninequalityholdsford-tuplesofcommutingcontractionswithd�2. B.K.Das Dilationofcommutingoperators Theorem(Nagy-Foias) LetTbeacontractiononaHilbertspaceH.ThenThasauniqueminimalunitarydilation. vonNeumanninequality:Foranypolynomialp2C[z],kp(T)ksupz2Djp(z)j: Theorem(T.Ando) Let(T1;T2)beapairofcommutingcontractionsonH.Then(T1;T2)dilates

toapairofcommutingunitaries(U1;U2). vonNeumanninequality:Foranypolynomialp2C[z1;z2],kp(T1;T2)ksup(z1;z2)2D2jp(z1;z2)j: NeitherdilationnorthevonNeumanninequalityholdsford-tuplesofcommutingcontractionswithd�2. B.K.Das Dilationofcommutingoperators Theorem(Nagy-Foias) LetTbeacontractiononaHilbertspaceH.ThenThasauniqueminimalunitarydilation. vonNeumanninequ

ality:Foranypolynomialp2C[z],kp(T)ksupz2Djp(z)j: Theorem(T.Ando) Let(T1;T2)beapairofcommutingcontractionsonH.Then(T1;T2)dilatestoapairofcommutingunitaries(U1;U2). vonNeumanninequality:Foranypolynomialp2C[z1;z2],kp(T1;T2)ksup(z1;z2)2D2jp(z1;z2)j: NeitherdilationnorthevonNeumanninequalityholdsford-tuplesofcommutingcontractionswithd�2. B.K.Das Dilationofc

ommutingoperators Theorem(Nagy-Foias) LetTbeacontractiononaHilbertspaceH.ThenThasauniqueminimalunitarydilation. vonNeumanninequality:Foranypolynomialp2C[z],kp(T)ksupz2Djp(z)j: Theorem(T.Ando) Let(T1;T2)beapairofcommutingcontractionsonH.Then(T1;T2)dilatestoapairofcommutingunitaries(U1;U2). vonNeumanninequality:Foranypolynomialp2C[z1;z2],kp(T1;T2)ksup(z1;z2)2D2

jp(z1;z2)j: NeitherdilationnorthevonNeumanninequalityholdsford-tuplesofcommutingcontractionswithd�2. B.K.Das Dilationofcommutingoperators ExplicitdilationandsharpvNinequality Theorem({&Sarkar,17) Let(T1;T2)beapairofcommutingcontractionsonHwithT1ispureanddimDTi1,i=1;2.Then(T1;T2)dilatesto(Mz;M)onH2DT1(D).Therefore,thereexistsavarietyV D2suchthatkp(T1;T2)k

;sup(z1;z2)2Vjp(z1;z2)j(p2C[z1;z2]):If,inaddition,T2ispurethenVcanbetakentobeadistinguishedvarietyofthebidisc. B.K.Das Dilationofcommutingoperators Dilationofaclassofcommutingoperators Letd�2and1pqd.Tdp;q=f(T1;:::;Td):^Tp;^TqsatisfySzegopositivityand^Tpispureg Theorem({,Barik,Haria&Sarkar,18) LetT=(T1;:::;Td)2Tdp;q.ThenTdilatesto(Mz1;:::;Mzp�1;Mp;Mz

p+1;:::;Mzq�1;Mq;Mzq;:::;Mzd�1);onH2E(Dd�1)withp(z)q(z)=q(z)p(z)=zpIE;forsomeHilbertspaceE. B.K.Das Dilationofcommutingoperators Dilationofaclassofcommutingoperators Letd�2and1pqd.Tdp;q=f(T1;:::;Td):^Tp;^TqsatisfySzegopositivityand^Tpispureg Theorem({,Barik,Haria&Sarkar,18) LetT=(T1;:::;Td)2Tdp;q.ThenTdilatesto(Mz1;:::;Mzp�1

;Mp;Mzp+1;:::;Mzq�1;Mq;Mzq;:::;Mzd�1);onH2E(Dd�1)withp(z)q(z)=q(z)p(z)=zpIE;forsomeHilbertspaceE. B.K.Das Dilationofcommutingoperators Problems Problem1:Characterized-tuplesofcommutingcontractionswhichadmitisometry/unitarydilations. Problem2:Findacharacterizationd-tuplesofcommutingcontractionswhichcanbedilatedtod-isometries. Problem3:Whatarethed-

tuplesofcommutingcontractionswhichsatisfyvNinequality? B.K.Das Dilationofcommutingoperators Problems Problem1:Characterized-tuplesofcommutingcontractionswhichadmitisometry/unitarydilations. Problem2:Findacharacterizationd-tuplesofcommutingcontractionswhichcanbedilatedtod-isometries. Problem3:Whatarethed-tuplesofcommutingcontractionswhichsatisfyvNinequality? B.K.Das Dilationofcom

mutingoperators Problems Problem1:Characterized-tuplesofcommutingcontractionswhichadmitisometry/unitarydilations. Problem2:Findacharacterizationd-tuplesofcommutingcontractionswhichcanbedilatedtod-isometries. Problem3:Whatarethed-tuplesofcommutingcontractionswhichsatisfyvNinequality? B.K.Das Dilationofcommutingoperators References J.AglerandJ.E.McCarthy,DistinguishedVarieties,Act

aMath.194(2005),133-153. B.K.DasandJ.Sarkar,Andodilations,vonNeumanninequality,anddistinguishedvarieties,JournalofFunctionalAnalysis272(2017),2114-2131. S.Barik,B.K.Das,K.J.HariaandJ.Sarkar,IsometricdilationsandvonNeumanninequalityforaclassoftuplesinthepolydisc,TransactionoftheAmericanMathematicalSociety(toappear). B.K.Das Dilationofcommutingoperators ThankYou B.K.Das Dilationof

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