第四章--谱与紧算子

11,XY(,)ABXY∈Axxyλ−=()AIxyλ−=0Axxλ−=()AIxλθ−=(,)ABXY∈(,)BBYX∈YABI=XBAI=1AB−=(,)ABXY∈(){}NAθ=()RAY=()ABX∈Rλ∈xX∈Axxλ=()AIxλθ−=λλ{}()()[0,1]|''()[0,1],(0)(1)DAxtCxtCxx=∈∈=:()[0,1]ADAC→()()''()Axtxt=−()()xtDA∈''''0Axxxxxxλλλ=⇔−=⇔+=()cossinxtatbtλλ=+,Rab∈2(2)nλπ≠nZ∈2(2)nλπ=nZ∈sin2ntπcos2ntπ2(2)nπ2[,]XCab=12(),(),,()nftftft12(),(),,()ngtgtgtX∈11()()[()()]()[()()]()nnbbkkkkaakkAxtftgsxsdsgsxsdsft====∑∑∫∫()xtX∈()ABX∈Cλ∈0Axxλ−=1[()()]()()0nbkkakftgsxsdsxtλ=−=∑∫1[()()]()()0nbkkakgsxsdsftxtλ=−=∑∫0λ=1[()()]()0nbkkakgsxsdsft==∑∫()xt0λ=0λ≠11()[()()]()nbkkakxtgsxsdsftλ==∑∫1()()nkkkxtdft==∑111()()()()nnnbkkllkkaklkftgsfsdsddftλ===⎡⎤=⎢⎥⎣⎦∑∑∑∫1[()()]nbkllkalgsfsdsddλ==∑∫1,2,,kn=()()bklklacgsfsds=∫()klCc=12(,,,)Tnddd=d()0Cλ−=dλλλ0x()AIxfλ−=()AIxfλ−=0xxg=+()gNAIλ∈−()ABX∈AIλ−λ3{}()AAρ=AIλ−λA()pAσ()()pAAσσ⊆{}AAσ()C()AAρσ=−{}()()ANAIλρλθ∈⇔−=()RAIXλ−=λ⇔(){}NAIλθ−=()RAIXλ−=()ABX∈||||1AIA−120()nnIAAIAA∞−=−==+++∑11||()||1||||IAA−−≤−1n20nknnkSAIAAA===+++=∑{}0nnS∞=()BX1mn()1111||||1||||||||||||||||||||1||||1||||nmnnmmkkmnknknAAASSAAAA+−+=+=+−−=≤=−−∑∑0→,mn→∞()BX()SBX∈||||0nnlimSS→∞−=0nnSA∞==∑1()()nnnIASIASIA+−=−=−1lim()limlim()nnnnnnIASIAISIA+→∞→∞→∞−=−==−IA−10()nnIASA∞−=−==∑101||||1||||lim||||lim||||lim1||||1||||nnknnnnkASSAAA+→∞→∞→∞=−=≤==−−∑{}Xθ≠()ABX∈4Cλ∈||||||Aλ()Aλρ∈110()()nnnARAAIλλλ∞−+==−=∑1||()||||||||RAAλλ≤−0()Aλρ∈001C|||()||()||ARAλλλλρ⎧⎫⎪⎪∈−⊂⎨⎬⎪⎪⎩⎭()Aσ≠∅0λ≠AAIIλλλ⎛⎞−=−−⎜⎟⎝⎠||||1AλAIλ−()11101nnnAAAIIλλλλ−∞−+=⎛⎞−=−−=−⎜⎟⎝⎠∑1||()||||||||RAAλλ≤−001||||()||RAλλλ−()()010000000()()()()()()AIAIIIAIIAAIIRAλλλλλλλλλλλλ−−=−+−=−−−−=−−−00|()()|1RAλλλ−0001110000()()(()())()()()nnnAIAIIRARARAλλλλλλλλλ∞−−−=−=−−−=−∑001C|||()||()||ARAλλλλρ⎧⎫⎪⎪∈−⊂⎨⎬⎪⎪⎩⎭()*()fBX∈||||1f≤()Aσ=∅1()(())(())zFzfRAfAzI−==−()CzAρ∈=()Fz1|()|||()||||||||zFzRAzA≤≤−||||||zA5||lim()0zFz→∞=()0zRA=()()0zIAIRAλ=−=()Aσ≠∅()ABX∈()max{|||()}rAAσλλσ=∈()rAσ()||||rAAσ≤()lim||||nnnrAAσ→∞=()ABX∈()pz{}(())()|()pApAσλλσ=∈()Aλσ∈()()pzpλ−()()zqzλ−()()()()pApIAIqAλλ−=−AIλ−()()()()AIqApApIλλ−=−()(())ppAλσ∈(())pAλσ∈12,,,,naλλλ12,,,,nlkkk11()()()nkknpzazzλλλ−=−−11()()()nkknpAIaAIAIλλλ−=−−()pAIλ−1,,nAIAIλλ−−iAIλ−1in≤≤()iAλσ∈()ipλλ=()(())ippAλλσ=∈(())(())pApAσσ⊂(())(())pApAσσ=62Hilbert()(,)()()baxtktsxsdsftλ−=∫()(,)()0baxtktsxsdsλ−=∫0λ≠f()(,)()0baxtkstxsdsλ−=∫(compactoperator)1fH∈2gH∈gf⊗1H2H()(,)gfxxfg⊗=1xH∀∈{}()|RgffFλλ⊗=∈gf⊗1H2H12:FHH→1H2HF{}1()|RFFxxH=∈()n+∞12:FHH→1122nnFgfgfgf=⊗+⊗++⊗()RF{}12,,,nggg1xH∈1122()()()nnFxaxgaxgaxg=+++(),1,2,,iaxFin∈=1,xyH∈Fλ∈()()()112211221122111222()()()()()()()()()()()()()()()()nnnnnnnnnFxyaxygaxygaxygFxFyaxgaxgaxgaygaygaygaxaygaxaygaxayg+=++++++=+=+++++++=++++++112222112222()()()()()()()FxaxgaxgaxgFxaxgaxgaxgλλλλλλλλ=+++==+++7(),1,2,,iaxin=1H122,,,nfffH∈1()(,),,1,2,,iiaxxfxHin=∈=1122nnFgfgfgf=⊗+⊗++⊗()n+∞F{}12,,,nggg()RF1,2,,()nfffNF⊥12||||||||||||1nfff====111222nnnFgfgfgfλλλ=⊗+⊗++⊗120nλλλ≥≥≥F()NF⊥()RFdim(())dim(())NFRFn⊥==12(,)KBHH∈{}121(,)nnFBHH+∞=⊂limnnFK→∞=lim||||0nnFK→∞−=KK0r()(,)KBrθ2H{}1(,)|||||BrxHxrθ=∈≤{}11nnxH∞=⊂||||,1,2,nxMn≤∞={}1knkx∞={}1knkKx∞=KH()ABH∈,AKKAKH*K{}1nnF∞=lim||||0nnKF→∞−=||||||||||||nnAKAFAKF−≤−||||||||||||,nnKAFAAKF−≤−,AKKA*||*||||||nnKFKF−=−*K2[,]XLab=(,)kst2[,][,]|(,)|ababkstdsdt×∞∫∫K()()(,)()baKxtkstxsds=∫2()[,]xtLab∈8222222|()()||(,)()|[|(,)||()|]|(,)||()|bbbaaabbbaaabbbaaaKxtdtkstxsdsdtkstdsxsdsdtkstdsdtdtxsds=≤=∫∫∫∫∫∫∫∫∫222[,][,]|||||(,)|||||ababKxkstdsdtx×≤∫∫1/22[,][,]|||||(,)|ababKkstdsdt×⎛⎞≤⎜⎟⎜⎟⎝⎠∫∫K2[,]Lab{}1()nntφ∞=2([,][,])Labab×,(,)()()nmnmtstsψφφ=,1nm≥22222,|||||()||()|||||||||1bbnmnmnmaatdtsdsψφφφφ===∫∫,1nm≥()()(),',''',,,(')(')nmnmnnmmnnmmψψφφφφδδ==−−,',,'1nnmm≥{},,1nmnmψ∞=2(,)([,][,])xtsLabab∈×,,1{}nmnmxψ∞=⊥()(),0,(,)()()(,)()()bbbbnmnmnmaaaaxxststdsdtxstsdstdtψφφφφ===∫∫∫∫,1nm≥'()(,)()bnaxtxstsdsφ=∫2'()[,]xtLab∈(',)0mxφ='mxφ⊥1m≥{}1()mmtφ∞='0x='()(,)()bnaxtxstsdsφ=∫{}1()nntφ∞=(,)0,..xstae≡[,][,]abab×0x={},,1nmnmψ∞=()2[,][,]Labab×,,,1(,)(,)nmnmnmkststλψ+∞==∑22,,1||(,)||||nmnmkstλ∞==∞∑1,N,,11(,)(,)NnmnmnNmMkststλψ≤≤≤≤=∑(),,1()()(,)()()()()NbbNnnmnmaanmKxtkstxsdssxsdstλφφ===∑∫∫2()([,])xtLab∈9NK2([,])LabN()()2222,,22,||()||||()()()|||||()()|||||||bbNnmnmnmnaanNmNmNnNnmmNnNKKxxsstxssxλφφλφλ∞∞∞−===∑∑∫∫∑1/22,,||||||0,NnmnmNKKNλ∞⎛⎞−≤→→∞⎜⎟⎝⎠∑K2(,)hLππ∈−()ht2π()()()()Kfxhxyfydyππ−=−∫2(,)fLππ∈−h()jntnnhthe∞=−∞=∑22||||||nnhh∞=−∞=∑()()()jnyjnxnnKfxhfyedyeππ∞−−=−∞=∑∫2(,)fLππ∈−()jmxjmxmKehe=mZ∈K{|}nhnZ∈(){|}{0}nKhnZσ=∈∪AλAIλ−HA{}()|pAAσλλ=AH()KBH∈0()Kσ∈K()pKλσ∈0λ≠()NKIλ−Cλ∈0λ≠λKK10K1KKI−=HIHKdim()NKIλ−=∞()NKIλ−{}1nne∞=nnKeeλ=1,2,n=K{}1nneλ∞='nn≠'||||||20nneeλλλ−={}1nneλ∞=dim()NKIλ−∞()pKλσ∉{}()NKIλθ−=KIλ−||||1inf||()||0xKIxλ=−||||1inf||()||0xKIxλ=−={}1nnx∞=||||1nx=||()||0,nKIxnλ−→→∞K1{}knkx∞={}knKx1()kkknnnxKIxKxλλ−⎡⎤=−−−⎣⎦1{}knkx∞=0limknkxx→∞=0||||1x=0()KIxλθ−=00Kxxλ=()pKλσ∉||||1inf||()||0xKIxλ=−||||1inf||()||0xKIxδλ==−()RKIλ−{}()NKIλθ−=()RKIHλ−=F||||||/2KFλ−()1()()KIKFIFKFIFλλλλ−−=−−+=−−+KFIλ−−()1()()KIKFIIKFIFλλλ−−=−−−−−1()FKFIFλλ−=−−Fλ()()KIKFIIFλλλ−=−−−KFIλ−−dim(())dim(())NKINIFλλ−=−*dim(())dim(())NIFNIFλλ−=−()()()()**0dim()dim()dim()dim()NKINIFNIFNKIλλλλ=−=−=−=−*()()()RKINKINKIHλλλ⊥⊥−=−=−=KIλ−11λKKH