Fredholm算子及其应用
毕业论文关键词 Fredholm 算子 积分算子 紧扰动 Fredholm 谱集 Fredholm 预解集 Sturm-Liouville 算子 谱分解
Title Fredholm Operator and its application
Abstract Fredholm Operator get its name from Fredholm and other mathematicians’s study of Integral Operators and Differential Operators,and it can links to the theory of index ,so it is an important subject.In this Article ,I study the space consist of Fredholm Oprator,The parametrix, Fredholm Alternatives via Dual Pairs, Applications to Integral Equations and Boundary-Value Problems.After that ,I study the maximal operator,minimal operator ,the self-adjoint operator corresponding to Sturm-Liouville differential expression,their Fredholm spectrum ,Fredholm resolvent. Keywords Fredholm Operator Integral Operator Compact Perturbation Fredholm spectrum Fredholm resolvent Sturm-Liouville Operator
Contents
1 Fredholméééfffƒƒƒ 555üüü 2
1.1 Fredholm½¬9ƒ 5ü 2
1.2 Fredholméf[ƒ ) 3
1.3 Fredholméf;6ƒ 4
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2.1 »©éf 5
2.2 3»©êß⁄>äØK•A^ 7
3 AAA^^^Sturm-LiouvilleéééfffÔÔÔƒƒƒ˛˛˛ 12
3.1 Sturm-LiouvilleéfÃx!é) 12
3.2 FredholmÃ8!Fredholm˝)8 20
4 (((ÿÿÿ 26
1 Fredholméf ƒ 5ü1.1 Fredholm ½¬9ƒ 5ü½¬µX!YèDâÇ5òmßTèX Y k.Ç5éfßÖß çIkÅßK°TèFredholméf"5µPT çIè (T),·Çk (T) =
(T)
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(T)èTéfºfòm ëÍ"e° ½nL²µ3k.éfUéfâÍ|§ òm•ßFredholméf|§ f8¥m "P“µïÏL£XßY§ß·Ç½¬F£X,Y§èlX Y §k Fredholméf|§ 8‹ßŸ•X,Y˛èBanachòm"½nµ S 2 F(X; Y )ßK3" > 0¶ T 2 L(X; Y )ÖjjT Sjj < "ûßkT 2 F(X; Y ); (T) = (S)y²µ-N := N(S);R := R(S),ƒk Eéf^ T(u; v) = Tu + v; 8u 2 M; v 2 Q:Ÿ•MßQ˛èX fòmÖ˜vX = N +M;X = R + QKéf^ T : M Q ! XÇ5ÖÎY.e°·Çƒky²^ SèV ߃kd^ S EꙥŸè˜ ßb ^ S(u; v) =0.KkSu + v = 0; u 2 M; v 2 Q;=Su 2 R; v 2 Q,œdv = 0; Su = 0;ddu = 0; v = 0ß^ Sè¸ "源]自{优尔·~论\文}网·www.youerw.com/ ·Ç ejj ^ T ^ Sjjø© ßKéf^ TèèV ßœdßejjT Sjjø© ßKéf^ TèV ßKdß ˜ ·Çå± òm ©)XeµX = T(M) + Qç?ò⁄ß·ÇkµN(T) NØ¢˛ßTu = 0; u 2 M —^ T(u; 0) = 0;œdu = 0; N(T) Nd˛™å µdimN(T) dimN < 1 qœècodimT(M) = dimQ = codimR·Ç¿JN òáfòmP¶ ke° Ü⁄/™µN = N(T) + PKßX = (N(T) + P) +M = N(T) + (P +M),œdT3P +M˛è¸ ß R(T) = T(P) + T(M); dimT(P) = dimP=X = T(P) + T(M) + U;U + R(T) = Xd±˛ ™f·Çå µcodimR = codimT(M) = codimR(T) + dimP;dimN = dimN(T) + dimPd˛™w, codimR(T); dimN(T)ëͲkÅß=TèFredholméf"øÖ (T) = dimN(T) codR(T) = (dimN dimP) (codimR dimP) = dimN codimR = (S)œd½n y"1.2 Fredholméf [ƒ )½nµ-AµX ! Yèk.Ç5éfߟ•X!YèBanachòmßKe° ¸áÿ„¥¬d µ1:éfAèFredholméf2:3k.éfPl; Pr : Y ! X⁄;éfCl : X ! X,Cr : Y ! Y¶ µPlA = I + ClAPr = I + Cr5µŸ•éfpl; pr©O âA Ü[ƒ )⁄m[ƒ )"y²µ1 ) 2 :©O X!Y fòmV⁄WßøÖkµX = N(A) + V; Y = R(A) +W œèAèFredholméfߧ±˛° òm©)¥å±â "-P : X ! N(A) Q : Y ! WÖP!QÎYÖè˜ "½¬k.éfXeµB : R(A) +W ! Xß /™XeµB(u + w) = A 10 u; 8u 2 R(A);w 2 WŸ•A0 : V ! R(A)L´A : X ! Y3‘˛ Åõ" éfA0èÇ5” "qœèBA = I P; AB = I Q:œèP(X)⁄Q(Y )èkÅëòmßéfP!Q;ß-Pr = Pl := B=å"2 ) 1 :éfI +Cl; I +Cr—¥çIè0 Fredholméf"qPlA = I +ClµN(A) N(I + Cl),œdµdimN(A) dimN(I + Cl) < 1,