SPECTRAL PROPERTIES OF THE OPERATOR OF RIESZ POTENTIAL TYPE

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 8, August 998, Pages 9 97 S SPECTRAL PROPERTIES OF THE OPERATOR OF RIESZ POTENTIAL TYPE MILUTIN R DOSTANIĆ Commuicated by Palle E T Jorgese Abstract For the operator of the Riesz potetial type we estimate the secod term i the asymptotic behavior of the spectrum ad fid its regularized trace Itroductio ad otatio For some classes of itegral operators there are methods of fidig the first term i the asymptotic of the sigular values or eigevalues see eg [], [3] However, fidig higher terms is rather difficult ad ca be realized oly i some situatios, ad depeds essetially o the structure of the kerel of a operator I this paper, we cosider the operator of Riesz potetial type A: L, L, defied by Afx = x y α fydy, <α< It occurs i the theory of Liouville fractioal itegrals see [] It is well kow see eg [8] that if <α<, the operator A is positive ad compact Deote by λ A, λ A, the eigevalues of A arraged i decreasig order, accordig to their multiplicity Alsoitiskow[],[3],[6],[9]that holds, where α λ A =cα cα =Γαcos α I what follows we deote by b α Lx, y dy the itegral operator actig o a L a, b, whose kerel is Lx, y For a compact operator T we deote by s T the-th eigevalue of the operator T =T T, ie s T =λ T Received by the editors Jauary 7, Mathematics Subject Classificatio Primary 47B Key words ad phrases Sigular values, eigevalues, regularized trace 9 c 998 America Mathematical Society Licese or copyright restrictios may apply to redistributio; see

2 9 MILUTIN R DOSTANIĆ By c p we deote the set of the compact operators T such that T p = s p p k T < k= I particular c is the set of uclear operators eigevalues of A it follows that A is ot uclear From the asymptotic of the Result Theorem a For every <r<the followig asymptotic formula holds: λ A = cα α +o r cα =Γαcos α α b The series = λ A cα coverges ad α λ A cα α = 4 α ζ α = holds, where ζ is the Reima zeta fuctio Proof of a For <α< defie the fuctio Hx, y = x y +4 α x+y+4+ α = ad the operator B : L, L,, Bfx = Hx, yfy dy Let ϕ x =si +x, =,,3,, ad let Kξ = e itξ t α dt =Γαcos α ξ α Observe that {ϕ } = is a orthoormal basis of L, By direct computatio oe fids that Hx, yϕ y dy = K ϕ x Therefore, the operator B is selfadjoit ad positive ad λ B = cα α From we obtai x y α = Hx, y+ x+y+ α + x+y α + x y 4 α x y +4 α x+y+4+ α, Licese or copyright restrictios may apply to redistributio; see

3 THE OPERATOR OF RIESZ POTENTIAL TYPE 93 By the result of Laptev [7], we obtai s x + y ± α dy = Oe c where c > is idepedet of O the other had, by the Birma-Solomjak theorem [], we have s W =Oe c where W deotes the operator with the kerel x y 4 α x y +4 α x+y+4+ α,, ad c > is idepedet of Hece, by the properties of the sigular values of the sum of two operators [5], from it follows that A = B + R where 3 s R c e c Fix r, <r< Choose θ so that r<θ< ad let k = k =[ θ ], m=m=[ θ ] Sice every atural umber ca be represeted as =k+m+j, j =,,,k, we obtai the estimate j = j θ + θ From the properties of the sigular values of the sum two operators it follows that s k+m+j B + R s km+j B+s m+ R, ie ie s B + R cα α s A cα α cα s km+jb cα α km + j α α + s m+r, + s m+ R Hece α+r s A cα α cα α+r km + j α 4 α + α+r s m+ R From the defiitio of the sequeces k, m adjadfrom3itfollows that α+r km + j α α = ad α+r s m+ R = Licese or copyright restrictios may apply to redistributio; see

4 94 MILUTIN R DOSTANIĆ Hece, by 4 α+r s A cα α Similarly, startig from s km+j B + R s k+m+j B s m+ R, we get α+r s A cα α ; thus α+r s A cα = Sice A>wehaves α A=λ Aad therefore λ A = cα α +o r Remark The above method does ot make it possible to aswer whether the followig formula holds: λ A = cα α +O Proof of b From a it follows that λ A = cα α +o r+α, hece, for r> α,theseries = λ A cα coverges Moreover for all α k =,, the series k cα λ k A α = is coverget For the proof of b we eed a lemma Lemma If C ad D are positive, compact operators o some complex Hilbert space such that C D Re z c < Re z<, thec z D z c ad C z D z si z si Re z C D Re z Proof If C D Re z c for some z, Re z<, the C D c ad hece CI + tc DI + td c for all t>iis the idetity operator Sice, for < Re z<, C z D z = si z t z CI + tc DI + td dt Licese or copyright restrictios may apply to redistributio; see

5 ad sice the fuctio s T Ado [], we have C z D z si z si z THE OPERATOR OF RIESZ POTENTIAL TYPE 95 = si z s +ts k= is operator mootoe, accordig to a result of t Re z CI + tc DI + td dt t Re z C D I + t C D dt s Re k z t Re z C D +t dt si z = si Re z C D Re z Applyig the lemma to the operators C = A, D = B we obtai from 3 that A B Re z c for every z, <Re z<, ad hece A z B z c If<Re z<, the applyig the lemma to C = A, D = B we obtai A z B z c,etc Therefore A z B z c for all z with Re z> I a similar way oe shows that the fuctio z tra z B z is aalytic for Re z> Cosider the fuctio Ψz = = λ z A cα α z From the asymptotic of λ A it follows that Ψz is aalytic for Re z > Re z> α, the both operators Az ad B z are uclear ad hece If tra z B z =tra z tr B z =Ψz Hece Ψz =tra z B z forrez> α It follows that Ψz =traz B z for all z, Rez>, ad hece k cα λ k A α =tra k B k, k =,, = It is especially simple to fid the first regularized trace of A: λ A cα α =tra B=trR = The itegral operator, o L,, with the cotiuous kerel x y 4 α x y +4 α x+y+4+ α, is uclear ad its trace is equal [5] to 4 α 4 α x+4+ α dx, Licese or copyright restrictios may apply to redistributio; see

6 96 MILUTIN R DOSTANIĆ The operators x + y + α dy ad x + y α dy are uclear ad, by [4], Theorem 3, it follows that Hece tr R = tr tr - x + y + α dy = x + y α dy = α 4α + 4 α After simplificatio we get This completes the proof of the theorem, tr R = 4 α ζ α x + α dx = 4α α, x + α dx = 4α α 4 α x+4+ α dx Remark A similar result ca be obtaied for the operator L: L, L, defied by Lfx = l x y fy dy It is kow [8] that L>adλ L= +O l Startig from the fuctio 5 kx = K x K is the McDoald fuctio, oe ca prove the formula λ L = + = C +l + t dt e 4+t where C is the Euler costat t Remark 3 By the same method oe ca fid the regularized trace of the covolutio operator with kerel of the form kx = x α L <α< x where L is a smooth fuctio such that x xl x Lx decreases for x large eough ad xl x = x Lx Licese or copyright restrictios may apply to redistributio; see

7 THE OPERATOR OF RIESZ POTENTIAL TYPE 97 Refereces [] T Ado, Compariso of orms fa fb ad f A B, Math Zeit , MR 9a:47 [] M Š Birma ad M Z Solomjak, Estimates of sigular values of the itegral operators, Uspekhi Mat Nauk 3, No , 7 84 MR 55:4 [3] M Š Birma ad M Z Solomjak, Asymptotic behaviour of the spectrum of weakly polar itegral operators, Izv Akad Nauk SSSR Ser Mat 34 97, No 5 MR 43:5359 [4] C Brislaw, Kerels of trace class operators, Proc Amer Math Soc 4 988, 8 9 MR 89d:4759 [5] I C Gohberg ad M G Krei, Itroductio to the theory of liear oselfadjoit operators, Traslatio of the math moographs, Vol 8, Amer Math Soc, Providece, RI 969 MR 39:7447 [6] M Kac, Distributio of eigevalues of certai itegral operators, Michiga Math J , 4 48 MR 9:7j [7] A Laptev, The spectral asymptotic of a class of itegral operators, Mat Zametki 6, No 5, 974, MR 5:8894 [8] J B Rid, Asymptotic behaviour of eigevalues of certai itegral equatios, Proceedigs of the Ediburgh Mat Soc 979, [9] M Roseblatt, Some results o the asymptotic behavior of eigevalues for a class of itegral equatios with traslatio kerels, J Math Mech 963, MR 7:547 [] S G Samko, A A Kilbas, ad O I Maricev, Fractioal itegrals ad derivatios ad some applicatios, Misk, 987 MR 89a:69 Matematicki Fakultet, Studetski trg6, Beograd, Yugoslavia Licese or copyright restrictios may apply to redistributio; see