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Hankel and Toeplitz operators

  • Jaak Peetre
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1573)

Keywords

Hardy Space Toeplitz Operator Singular Integral Equation Pseudodifferential Operator Bergman Space 
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References

  1. 1.
    Clark D. N., On interpolating sequences and the theory of Hankel and Toeplitz matrices, J. Functional Anal. 5 (1970), 247–258.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Khruščëv S. V., Nikol'skii N. K., Pavlov B. S., Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory. Seminar, Leningrad 1979/80, Lect. Notes Math., vol. 864, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp. 214–335.Google Scholar
  3. 3.
    Power S. C., Hankel Operators on Hilbert Space, Research Notes in Mathematics 64, Pitman, London, 1982.zbMATHGoogle Scholar

Reference

  1. 4.
    Megretskii A. V., A quasinilpotent Hankel operator, Algebra i Analiz 2 (1990), no. 4, 201–212 (Russian); English transl. in Leningrad Math. J. 2 (1991), no. 4.MathSciNetGoogle Scholar

References

  1. 1.
    Peller V. V., Estimates of functions of power bounded operators on Hilbert spaces, J. Oper. Theory 7 (1982), 341–372.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Varopoulos N. Th., Some remarks on Q-algebras, Ann. Inst. Fourier (Grenoble) 22 (1972), 1–11.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Varopoulos N. Th., Sur les quotiens des algèbres uniformes, C. R. Acad. Sci. Paris 274 (1972), 1344–1346.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Charpentier P., Q-algèbres et produits tensoriels topologiques, Thèse, Orsay, 1973.Google Scholar
  5. 5.
    Halmos P., Ten problems in Hilbert space, Bull. Am. Math. Soc. 76 (1970), 887–933.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Sarason D., Function theory on the unit circle, Notes for lectures at Virginia Polytechnic Inst. and State Univ., Blacksburg, Va., 1978.Google Scholar
  7. 7v.
    Fefferman Ch., Stein E. M., H p spaces of several variables, Acta Math. 129 (1972), 137–193.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8ac.
    Rochberg R., A Hankel type operator arising in deformation theory, Proc. Sympos. Pure. Math. 35 (1979), no. 1, 457–458.zbMATHCrossRefGoogle Scholar
  9. 9w.
    Tonge A. M., Banach algebra and absolutely summing operators, Math. Proc. Camb. Philos. Soc. 80 (1976), 465–473.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lindenstrauss J., Pełczyński A., Contribution to the theory of classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249.zbMATHCrossRefGoogle Scholar
  11. 11.
    Lindenstrauss J., Pełczyński A., Absolutely summing operators in L p-spaces and their applications, Studia Math. 29 (1968), 275–326.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Shields A. L., On Möbius bounded operators, Acta Sci. Math. 40 (1978), 371–374.MathSciNetzbMATHGoogle Scholar
  13. 13.
    van Casteren J. A., Operators similar to unitary and self-adjoint ones, Pacif. J. Math. 104 (1983), 241–255.zbMATHCrossRefGoogle Scholar

Reference

  1. 14.
    Paulsen V. I., Toward a theory of K-spectral sets, Surveys of some recent results in operator theory (J. Conway and B. Morrel, ed.), Pitnam Res. Math. Series, no. 171, Longman, UK, 1988, pp. 221–240.Google Scholar

References

  1. 1.
    Arazy J., Fisher S., Peetre J., Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989–1053.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arazy J., Fisher S., Janson S., Peetre J., Membership of Hankel operators on the ball in unitary ideals, Proc. London Math. Soc. 43 (1991), 485–508.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bergh J., Löfström J., Interpolation Spaces, Springer-Verlag, 1976.Google Scholar
  4. 4.
    Rochberg R., Feldman M., Singular value estimates for commutators and Hankel operators on the unit ball and the Heisenberg group, Analysis and PDE, A collection of papers dedicated to Mischa Cotlar (Cora Sadosky, ed.), Marcel Dekker Inc., 1990, pp. 121–160.Google Scholar
  5. 5.
    Nikol'ski i N. K., Treatise on the shift operator, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986.CrossRefGoogle Scholar
  6. 6.
    Nowak K., Weak type estimates for singular values of commutators on weighted Bergman spaces, Indiana Univ. Math. J. (to appear).Google Scholar
  7. 7w.
    Peller V. V., Hankel operators of class Open image in new windowand their applications, Mat. Sb. 113 (1980), 538–581 (Russian); English transl. in Math. USSR, Sb. 41 (1982), 443–479.MathSciNetzbMATHGoogle Scholar
  8. 8ad.
    Peller V. V., A description of Hankel operators of class Open image in new windowfor p>0, an investigation of the rate of rational approximation, and other applications, Mat. Sb. 122 (1983), 481–510 (Russian); English transl. in Math. USSR, Sb. 50 (1985), 465–494.MathSciNetGoogle Scholar
  9. 9x.
    Rochberg R., Semmes S., Nearly weakly orthonormal sequences, singular value estimates and Calderón-Zygmund operators, J. Funct. Anal. 86 (1989), 237–306.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Rochberg R., Semmes S., End point results for estimates of singular values of integral operators., Contribution to operator theory and its applications (Gohberg et al, eds.), OT 35, Operator Theory: Advances and Applications, Birkhäuser, Basel-Boston-Berlin, 1988, pp. 217–231.CrossRefGoogle Scholar

References

  1. 1.
    [AAK] Adamyan V. M., Arov D. Z., Kre in M. G., Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb., Nov. Ser. 86 (1971), 34–75 (Russian); English transl. in Math. USSR Sbornik 15 (1971), 31–73.MathSciNetGoogle Scholar
  2. 2.
    [B] Baker H. F., Abel's Theorem and the Allied Theory Including the Theory of Theta Functions, Cambridge University Press, Cambridge, 1897.zbMATHGoogle Scholar
  3. 3.
    [GR] Gauchman H., Rubel L., Sums of products of functions of x times functions of y, Linear Algebra Appl. 125 (1989), 19–63.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    [J] Janson S., manuscript in preparation 1991.Google Scholar
  5. 5.
    [JPR] Janson S., Peetre J., Rochberg R., Hankel forms and the Fock space, Rev. Mat. Iberoamer. 3 (1987), 61–138.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    [JPW] Janson S., Peetre J., Wallstén R., A new look on Hankel forms over Fock space, Studia Math. 94 (1989), 33–41.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    [K] Kronecker L., Zur Theorie der Elimination einer Variablen aus zwei algebraischen Gleichungen, Monatsber. Königl. Preuss. Akad. Wiss. (1881), 535–600.Google Scholar
  8. 8.
    [M] Mumford D., Tata Lectures on Theta I, II, Birkhäuser, Basel, 1983.zbMATHCrossRefGoogle Scholar
  9. 9.
    [P] Pisier G., Interpolation between H p spaces and non-commutative generalizations I, preprint, 1990.Google Scholar

Reference

  1. 10.
    [MPT] Megretskii A., Peller V., Treil S., The inverse spectral problem for self-adjoint Hankel operators, Acta Math. (to appear).Google Scholar

References

  1. 1.
    Arazy J., Fisher S., Janson S., Peetre J., Membership of Hankel operators on the ball in unitary ideals, J. London Math. Soc. 43 (1991), 485–508.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Békollé D., Berger C., Coburn L., Zhu K., BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (1990), 310–320.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Li H., BMO and Hankel operators on multiply connected domains, preprint, 1990.Google Scholar
  4. 4.
    Li H., BMO and Hankel operators on strictly convex domains, preprint, 1990.Google Scholar
  5. 5.
    Zheng D., Schatten class Hankel operators on the Bergman space of bounded symmetric domains, Integral Equations Oper. Theory 13 (1990), 442–459.zbMATHCrossRefGoogle Scholar
  6. 6.
    Zhu K., Schatten class Hankel operators on the Bergman space of the unit ball, Am. J. Math. 113 (1991), 147–167.MathSciNetzbMATHCrossRefGoogle Scholar

References

  1. 1.
    Fefferman C., The multiplier problem for the ball, Ann. Math. 94 (1971), 330–336.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Peng L., Hankel operators on the Paley-Wiener space in disk, Proc. Cent. Math. Anal. Austral. Natl. Univ. 16 (1988), 173–180.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Rochberg R., Toeplitz and Hankel operators on the Paley-Wiener space, Integral Equations Oper. Theory 10 (1987), 186–235.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Taylor M., Pseudodifferential Operators, Princeton University Press, Princeton, N. J., 1981.zbMATHGoogle Scholar

References

  1. 1.
    Sarason D., Function Theory on the Unit Circle, Notes for Lectures at Virginia Polytechnic Inst. and State Univ., Blacksburg, Va., 1978.Google Scholar
  2. 2.
    Clark D. N., On a similarity theory for rational Toeplitz operators, J. Reine Angew. Math. 320 (1980), 6–31.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Wolff T., Two algebras of bounded functions, Duke Math. J. 49 (1982), 321–328.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Rosenblum M., The absolute continuity of Toeplitz's matrices, Pacif. J. Math. 10 (1960), 987–996.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Peller V. V., Invariant subspaces for Toeplitz operators, Zapiski Nauch. Semin. LOMI 126 (1983), 170–179 (Russian); English transl. in J. Soviet Math. 27 (1984), 2533–2539.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Sz.-Nagy B., Foiaş C. Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, 1970.zbMATHGoogle Scholar

References

  1. 7.
    Yakubovich D. V., Riemann surface models of Toeplitz operators, Toeplitz Operators and Spectral Function Theory, OT: Adv. Appl., vol. 42, Birkhäuser Verlag, Basel-Boston-Berlin, 1989.zbMATHGoogle Scholar
  2. 8.
    Atzmon A. Power regular operator (to appear).Google Scholar

References

  1. 1.
    Widom H., On the spectrum of a Toeplitz operator, Pacif. J. Math. 14 (1964), 365–375.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Douglas R. G., Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.zbMATHGoogle Scholar
  3. 3.
    Douglas R. G., Banach Algebra Techniques in the Theory of Toeplitz Operators, conference Board of the Math. Sciences, Regional Conf. Series in Math. 15, Am. Math. Soc., Providence, R. I., 1973.zbMATHCrossRefGoogle Scholar
  4. 4.
    Douglas R. G., Local Toeplitz operators, Proc. London Math. Soc. 36 (1978), 243–272.MathSciNetzbMATHCrossRefGoogle Scholar

References

  1. 5.
    Böttcher A., Silbermann B., Analysis of Toeplitz operators, Akademie Verlag, Berlin, 1989.zbMATHGoogle Scholar

References

  1. 1.
    Axler S., Chang S.-Y. A., Sarason D., Products of Toeplitz operators, Integral Equations Oper. Theory 1 (1978), 285–309.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Garnett J. B., Bounded Analytic Functions (1981), Academic Press, New York.zbMATHGoogle Scholar
  3. 3.
    Jones P. W., Estimates for the corona problem, J. Funct. Anal. 39 (1980), 162–181.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Sarason D., Function Theory on the Unit Circle, Notes for lectures at Virginia Polytechnic Inst. and State Univ., Blacksburg, Va., 1978.Google Scholar
  5. 5.
    Sarason D., The Shilov and Bishop decompositions of H +C, Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. II, Wadsworth, Belmont, Calif., 1983, pp. 461–474.zbMATHGoogle Scholar
  6. 6.
    Sarason D., Exposed points in H 1, II: Oper. Theory. Adv. Appl. 48 (1990), 333–347.MathSciNetzbMATHGoogle Scholar
  7. 7x.
    Volberg A. L., Two remarks concerning the theorem of S. Axler, S.-Y. A.Chang and D. Sarason, J. Oper. Theory 8 (1982), 209–218.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Wolff T. H., Some theorems on vanishing mean oscillation, Dissertation, University of California, Berkeley, Calif., 1979.Google Scholar

References

  1. 1.
    McDonald G., Sundberg C., Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), 595–611.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Widom H., On the spectrum of Toeplitz operators, Pacific J. Math. 14 (1964), 365–375.MathSciNetzbMATHCrossRefGoogle Scholar

References

  1. 1.
    Rabindranathan M., On the inversion of Toeplitz operators, J. Math. Mech. 19 (1969/70), 195–206.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Helson H., Szegö G., A problem in prediction theory, Ann. Mat. Pure Appl. 51 (1960), 107–138.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Devinatz A., Toeplitz operators on H 2 space, Trans. Am. Math. Soc. 112 (1964), 304–317.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Pousson H. R., Systems of Toeplitz operators on H 2. Trans. Am. Math. Soc. 133 (1968), 527–536.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Verbitskiî I. È., Krupnik N. Ya., Exact constants in theorems about boundedness of singular operators in weighted spaces, Linear operators, Shtiintsa, Kishinev, 1980, pp. 21–35. (Russian)Google Scholar
  6. 6.
    Simonenko I. B., Riemann's boundary problem for a pair of functions with measurable coefficients and its application to singular integrals in weighted spaces, Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 277–306 (Russian)MathSciNetGoogle Scholar
  7. 7.
    Krupnik N. Ya., Some consequences of the theorem of Hunt-Muckenhoupt-Wheeden, Operators in Banach spaces, Shtiintsa, Kishinev, 1978, pp. 64–70. (Russian)zbMATHGoogle Scholar
  8. 8ae.
    Spitkovskiî I. M., On factorization of matrix functions whose Hausdorff set lies inside an angle, Soobshch. Akad. Nauk Gruz. SSR 86 (1977), 561–564. (Russian)MathSciNetzbMATHGoogle Scholar
  9. 9y.
    Hunt R., Muckenhoupt B., Wheeden R., Weighted norm inequalities for conjugate function and Hilbert transform, Trans. Am. Math. Soc. 176 (1973), 227–251.MathSciNetzbMATHCrossRefGoogle Scholar

References

  1. 1.
    Sarason D., Toeplitz operators with semi-almost periodic symbols, Duke Math. J. 44 (1977), 357–364.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Saginashvili A. I., Singular integral equations with coefficients having discontinuities of semi-almost periodic type, Theory of analytic functions and harmonic analysis, Tr. Tbilis. Mat. Inst. Razmadze 66 (1980), 84–95. (Russian)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Karlovich Yu. I., Spitkovskiî I. M., On Noetherness of some singular integral operators with matrix coefficients of class SAP and systems of convolution equations on a finite interval associated with them, Dokl. Akad. Nauk SSSR 269 (1983), 531–535 (Russian); English transl. in Sov. Math. Dokl. 27 (1983), 358–263.MathSciNetGoogle Scholar
  4. 4.
    Karlovich Yu. I., Spitkovskiî I. M., On Noetherness, n-and d-normality of singular integral operators having discontinuities of semi-almost periodic type, School of theory of operators in functional spaces, abstracts, Minsk, 1982, pp. 81–82. (Russian)Google Scholar
  5. 5.
    Chebotarev G. N., Partial indices of the Riemann boundary problem with a triangular matrix of second order, Uspekhi Mat. Nauk 11 (1956), no. 3, 199–202. (Russian)MathSciNetGoogle Scholar

References

  1. 1.
    Gohberg I. C. and Kreîn M. G., The Theory of Volterra Operators in Hilbert Space and its Applications, Nauka, Moscow, 1967 (Russian); English translation: Translations of Math. Monographs, Am. Math. Soc., Providence, Rhode Island, 1970.zbMATHGoogle Scholar
  2. 2.
    Sakhnovich L. A., Factorizations of operators in L 2 (a, b), Funkts. Anal. i Prilozh. 13 (1979), no. 3, 40–45 (Russian); English transl. in Funct. Anal. Appl. 13 (1979), 187–192.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Sakhnovich L. A., On an integral equation with kernel depending on the difference of the arguments, Mat. Issled. 8 (1973), 138–146. (Russian)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kreîn M. G., Continuous analogues of propositions about polynomials orthogonal on the unit circle, Dokl. Akad. Nauk SSSR 106 (1955), 637–640. (Russian)Google Scholar
  5. 5.
    Sakhnovich L. A., On the factorization of an operator valued transfer operator function, Dokl. Akad. Nauk SSSR 226 (1976), 781–784 (Russian); English transl. in Sov. Math. Dokl. 17 (1976), 203–207.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Livshits M. S., Operators, Oscillations, Waves. Open Systems, Nauka, Moscow, 1966 (Russian); English translation: Translations of Math. Monographs, Am. Math. Soc., Providence, Rhode Island, 1973.Google Scholar
  7. 7y.
    Potapov V. P., On the multiplicative structure of J-non-expanding matrix functions, Tr. Mosk. Mat. O-va 4 (1955), 125–136 (Russian); English translation in Translations of Math. Monographs 15, Am. Math. Soc., Providence, Rhode Island, 1970, 131–243.Google Scholar

References

  1. 8af.
    Larsson D. R., Nest algebras and similarity transformations, Ann. Math., II. Ser. 121 (1985), 409–427.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 9z.
    Sakhnovich L. A., Factorization of operators in L 2 (a, b), Funkts. Anal. i Prilozh. 13 (1979), no. 3, 40–45 (Russian); English transl. in Funct. Anal. Appl. 13 (1979), 187–192.MathSciNetCrossRefGoogle Scholar

References

  1. 1.
    Coburn L. A., The C *-algebra generated by an isometry I, II, Bull. Amer. Math. Soc. 73 (1967), 722–726; Trans. Amer. Math. Soc. 137 (1969), 211–217.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Coburn L. A., Douglas R. G., Singer I. M., An index theorem for Wiener-Hopf operators on the discrete quarter-plane, J. Diff. Geom. 6 (1972), 587–593.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Douglas R. G., Howe R., On the C *-algebra of Toeplitz operators on the quarter-plane, Trans. Am. Math. Soc. 158 (1971), 203–217.MathSciNetzbMATHGoogle Scholar

References

  1. 1.
    Berger C. A., Coburn L. A., Toeplitz operators and quantum mechanics, J. Funct. Anal. 68 (1986), 273–299.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Berger, C. A., Coburn L. A., Toeplitz operators on the Segal-Bargmann space, Trans. Am. Math. Soc. 301 (1987), 813–829.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Guillemin V., Toeplitz operators in n-dimensions, Integral Equations Oper. Theory 7 (1984), 145–205.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Janas J., Toeplitz and Hankel operators in Bargmann space, Glasg. Math. J. 30 (1988), 315–323.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Janas J., Unbounded Toeplitz operators in the Bargmann-Segal space, Studia Math. (to appear).Google Scholar
  6. 6.
    Shapiro H. S., An algebraic theorem of E. Fisher, and the holomorphic Goursat problem, Bull. London Math. Soc. 21 (1989), 513–537.MathSciNetzbMATHCrossRefGoogle Scholar

References

  1. 1.
    Golinskiî B. L., Ibragimov I. A., On the Szegö limit theorem, Izv. Akad. Nauk SSSR 35 (1972), no. 2, 408–427. (Russian)zbMATHGoogle Scholar
  2. 2.
    Kreîn M. G., Spitkovskiî I. M., On factorizations of α-sectorial matrix functions on the unit circle, Mat. Issled. 47 (1978), 41–63 (Russian)Google Scholar
  3. 3.
    Kreîn M. G., Spitkovskiî I. M., Some generalizations of the first Szegö limit theorem, Anal. Math., 9 (1983), 23–41. (Russian)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kreîn M. G., Spitkovskiî I. M., On factorizations of matrix functions in the unit circle, Dokl. Akad. Nauk SSSR 234 (1977), 287–290 (Russian); English translation in Soviet Math. Dokl. 18 (1977), 287–299.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Devinatz A., The strong Szegö limit theorem, Illinois J. Math. 11 (1967), 160–175.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Basor E., Helton J. W., A new proof of the Szegö limit theorem and new results for Toeplitz operators with discontinuous symbol, J. Oper. Theory 3 (1980), 23–39.MathSciNetzbMATHGoogle Scholar
  7. 7z.
    Kreîn M. G., On an extrapolation problem by A. N. Kolmogorov, Doklady Akad. Nauk SSSR 46 (1945), no. 8, 306–309. (Russian)zbMATHGoogle Scholar
  8. 8ag.
    Mikaelyan L. V., Continuous matrix analogues of G. Szegö's theorem on Toeplitz determinants, Izv. Akad. Nauk Arm. SSR 17 (1982), 239–263 (Russian)MathSciNetzbMATHGoogle Scholar

References

  1. 9.
    Böttcher A., Silbermann B., Analysis of Toeplitz operators, Akademie-Verlag, Berlin, 1989.zbMATHGoogle Scholar
  2. 10.
    Gohberg I., Kaashoek M. A., Asymptotic formulas of Szegö-Kac-Achiezer type., Asymptotic Analysis 5 (1992), no. 3, 187–220.MathSciNetzbMATHGoogle Scholar

References

  1. 1.
    Barnsley, Bessis D., Moussa P., The Diophantine moment problem and the analytic structure in the activity of the ferromagnetic Ising model, J. Math. Phys. 20 (1979), 535–552.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Vladimirov V. S., Volovich I. V., The Ising model with a magnetic field and a Diofantine moment problem, Teor. Mat. Fiz. 53 (1982), no. 1, 3–15 (Russian); English transl. in Theor. Math. Phys. 50 (1982), 177–185.MathSciNetGoogle Scholar
  3. 3.
    Helson H., Note on harmonic functions, Proc. Am. Math. Soc. 4 (1953), 686–691.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Vladimirov V. S., Volovich I. V., On a model in statistical physics, Teor. Mat. Fiz. 54 (1983), no. 1, 8–22 (Russian); English transl. in Theor. Math. Phys. 54 (1983), 1–11.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Vladimirov V. S., Volovich I. V., The Wiener-Hopf equation, the Riemann-Hilbert problem and orthogonal polynomials, Dokl. Akad. Nauk SSSR 266 (1982), 788–792 (Russian); English transl. in Soviet Math. Doklady 26 (1982), 415–419.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Szegö G., Orthogonal Polynomials, A. M. S. Coll. Publ. 23, revised ed., Am. Math. Soc., New York, 1959.zbMATHGoogle Scholar
  7. 7aa.
    Golinskiî B. L., Asymptotic representation of orthogonal polynomials, Uspekhi Mat. Nauk 35 (1980), no. 2, 145–196. (Russian)MathSciNetGoogle Scholar
  8. 8ah.
    Linnik I. Yu., A multidimensional analogue of Szegö's theorem, Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 1393–1403 (Russian); English transl. in Math. USSSR, Izv. 9 (1975), 1323–1332.MathSciNetGoogle Scholar

References

  1. 9aa.
    Sakhnovich A. L., On a class of extremal problems, Izv. Akad. Nauk SSSR, Ser. Mat. 51 (1987), 436–443 (Russian); English transl. in Math. USSR, Izv. 30 (1988), 411–418.MathSciNetzbMATHGoogle Scholar
  2. 10.
    Sakhnovich A. L., On Toeplitz block matrices and connected properties of the Gauss model on a halfaxis, Teor. Mat. Fiz. 63 (1985), no. 1, 154–160 (Russian); English transl. in Theor. Math. Phys. 63 (1985), 27–33.MathSciNetCrossRefGoogle Scholar
  3. 11.
    Helson H., Sz.-Nagy B., Vasilescu F. H. (ed.), Linear Operators in Function Spaces, Birkhäuser, Basel, 1990.Google Scholar

References

  1. 1.
    Axler S., Chang S. Y. A., Sarason D., Products of Toeplitz operators, Integral Equations Oper. Theory 1 (1978), 285–309.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Böttcher A., Silbermann B. Analysis of Toeplitz Operators, Akademie-Verlag, 1989 and Springer-Verlag, 1990.Google Scholar
  3. 3.
    Havin V. P., Wolf H., The Poisson kernel is the only approximate identity that is asymptotically multiplicative on H , Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklova 170 (1989), 82–89. (Russian)Google Scholar
  4. 4.
    Nikol'skiî N. K., Treatise on the Shift Operator, Grundlehren 273, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986.CrossRefGoogle Scholar
  5. 5.
    Volberg A. L., Two remarks concerning the theorem of S. Axler, S.-Y. A. Chang and D. Sarason, J. Oper. Theory 7 (1982), 209–218.MathSciNetzbMATHGoogle Scholar

Literatur

  1. 1.
    Arnold D. N., Wendland W. L., On the asymptotic convergence of collocation methods, Math. Comput. 41 (1983), 349–381.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Dang D. Q., Norrie D. H., A finite element method for the solution of singular integral equations, Comput. Math. Appl. 4 (1978), 219–224.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Elschner J., Prössdorf S., Über die starke Finite element methods for singular integral equations on an interval, Engineering Analysis 1 (1984), 83–87.CrossRefGoogle Scholar
  4. 4.
    Gohberg I. C., Fel'dman I. A., Convolution Equations and Projection Method for their Solutions, Nauka, Moscow, 1971 (Russian); English translation: Amer. Math. Soc., Providence, R. I., 1974zbMATHGoogle Scholar
  5. 5.
    Gohberg I. C., Krupnik N. Ya., Singular integral operators with piecewise continuous coefficients and their symbols, Izv. Akad. Nauk SSSR 35 (1971), 940–961 (Russian) English translation in Soviet Math., Izv. 5 (1971), 955–979.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gohberg I. C., Krupnik N. Ya., Introduction to the Theory of one Dimensional Singular Integral Operators, Shtiintsa, Kishinev, 1973 (Russian); German translation: Birkhäuser, Basel, 1979Google Scholar
  7. 7ab.
    Ien E., Srivastav R. P., Cubic splines and approximate solution of singular integral equations, Math. Comput. 37 (1981), 417–423.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8ai.
    Kohn I. I., Nirenberg L. I., An algebra of pseudodifferential operators, Commun. Pure and Appl. Math. 18 (1965), 269–305.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Michlin S. G., Prössdorf S., Singuläre Integraloperatoren, Akademie-Verlag, Berlin, 1980; English translation; Springer-Verlag, Berlin et al, 1986zbMATHGoogle Scholar
  10. 10.
    Prössdorf S. Zur Splinekollokation für lineare Operatoren in Sobolewräumen, Recent trends in mathematics, Teubner-Texte zur Math. 50, Teubner, Leipzig, 1983, pp. 251–262.zbMATHGoogle Scholar
  11. 11.
    Prössdorf S., Schmidt G., A finite element collocation method for singular integral equations, Math. Nachr. 100 (1981), 33–60.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Prössdorf S., Rathsfeld A., A spline collocation method for singular integral equations with piecewise continuous coefficients, Integral Equations Oper. Theory 7 (1984), 536–560.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Schmidt G., On spline collocation for singular integral equations, Math Nachr. 111 (1982), 177–196.MathSciNetzbMATHCrossRefGoogle Scholar

Supplement zur Literatur

  1. 1.
    1*. Elschner J., On spline approximation for singular integral equations on an interval, Math. Nachr. 139 (1988), 309–319.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    2*. Prössdorf S., Rathsfeld A., On strongly elliptic singular integral operators with piecewise continuous coefficients, Integral Equations Oper. Theory 8 (1985), 825–841.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    3*. Prössdorf S., Rathsfeld A., Mellin techniques in the numerical analysis for onedimensional singular integral operators, Report R-MATH-06/88, Karl-Weierstraß-Inst. Math. Akad. Wiss. DDR, Berlin, 1988.zbMATHGoogle Scholar
  4. 4.
    4*. Prössdorf S., Silbermann B., Numerical Analysis for Integral and Related Operator Equations, Akademie-Verlag, Berlin, 1991, and Birkhäuser, Basel-Boston-Berlin, 1991.zbMATHGoogle Scholar
  5. 5.
    5*. Saranen J., Wendland W. L., On the asymptotic convergence of collocation methods with spline functions of even degree, Math. Comput. 45 (1985), 91–108.MathSciNetzbMATHCrossRefGoogle Scholar

References

  1. 1.
    Vekua N. P., Systems of Singular Integral Equations and Some Boundary Problems, 2nd revised ed., Nauka, Moscow, 1970 (Russian); English translation (of 1st ed.): Noordhoff, Groningen, 1967Google Scholar
  2. 2.
    Vekua N. P., Generalized Analytic Functions, Fizmatgiz., Moscow (1959) (Russian); English translation: Pergamon Press, London-Paris-Frankfurt, Addison Wesley, Reading, Mass., 1962.zbMATHGoogle Scholar
  3. 3.
    Litvinchuk G. S., Boundary Problems and Singular Integral Equations with Shift, Nauka, Moscow, 1977. (Russian)zbMATHGoogle Scholar
  4. 4.
    Spitkovskiî I. M., On the theory of generalized Riemann boundary value problem in L p classes, Ukr. Mat. Zh. 31 (1979), no. 1, 63–73 (Russian); English translation in Ukr. Math. J. 31 (1979), 47–57.CrossRefGoogle Scholar
  5. 5.
    Spitkovskiî, I. M., On multipliers not affecting factorizability, Dokl. Akad. Nauk SSSR 231 (1976), 1300–1303 (Russian); English translation in Soviet Math. Doklady 17 (1976), 1733–1739.MathSciNetGoogle Scholar
  6. 6.
    Litvinchuk G. S., Spitkovskiî I. M., Sharp estimates for the defect numbers of a generalized Riemann boundary value problem, factorization of Hermitian matrix-valued functions and some problems of approximation by meromorphic functions, Mat. Sb. Nov. Ser. 117 (1982), 196–214 (Russian); English translation in Math. USSR, Sb. 45 (1983), 205–224.Google Scholar
  7. 7ac.
    Adamyan V. M., Arov D. Z., Kreîn M. G., Infinite Hankel matrices and generalizations of the problems Carathéodory-Fejér and F. Riesz, Funkts. Anal. i Prilozh. 2 (1968), no. 1, 1–19 (Russian); English translation in Funct. Anal. Appl. 2 (1968), 1–20.zbMATHCrossRefGoogle Scholar
  8. 8aj.
    Adamyan V. M., Arov D. Z., Kreîn M. G., Infinite Hankel matrices and generalizations of the problems of Carathéodory-Fejér and I. Schur, Funkts. Anal. i Prilozh. 2 (1968), no. 4, 1–17 (Russian); English translation in Funct. Anal. Appl. 2 (1968), 269–281.CrossRefGoogle Scholar
  9. 9ab.
    Adamyan V. M., Arov D. Z., Kreîn M. G., Analytic properties of the Schmidt pair of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. 86 (1971), 33–73 (Russian); English translation in Math. USSR Sbornik 15 (1971), 31–73.Google Scholar
  10. 10.
    Zverovich È. I., Litvinchuk G. S., Onesided boundary problems in the theory of analytic functions, Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 1003–1036 (Russian)MathSciNetGoogle Scholar
  11. 11.
    Boyarskiî B. V. (=Bojarski, B.), Analysis of the solvability of boundary problems in function theory, Studies in contemporary problems in the theory of functions of one complex variable, Fizmatgiz, Moscow, 1961, pp. 57–79. (Russian)Google Scholar

References

  1. 1.
    Semënov-Tyan-Shanskiį M. A., What is a classical τ-matrix?, Funkts. Anal. i Prilozh. 17 (1983), no. 4, 17–33 (Russian); English transl. in Funct. Anal. Appl. 17 (1983), 259–272.Google Scholar
  2. 2.
    Belavin A. A., Drinfel'd V. G., On the solutions of the classical Yang-Baxter equation for simple Lie algebras, Funkts. Anal. i Prilozh. 16 (1982), no. 3, 1–29 (Russian); English transl. in Funct. Anal. Appl. 16 (1982), 159–180.MathSciNetzbMATHGoogle Scholar

Reference

  1. 1.
    Coifman R. R., Rochberg R., Weiss G., Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635.MathSciNetzbMATHCrossRefGoogle Scholar

References

  1. 2.
    Arazy J., Fisher S., Peetre J., Hankel operators on weighted Bergman spaces, Am. J. Math. 110 (1988), 989–1054.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 3.
    Axler S., The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), 315–332.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 4.
    Stroenthoff K., Compact Hankel operators on the Bergman space, Ill. J. Math. 34 (1990), 159–174.MathSciNetGoogle Scholar
  4. 5.
    Zhu K., VMO, ESV, and Toeplitz operators on the Bergman space, Trans. Amer. Math. Soc. 302 (1987), 617–646.MathSciNetCrossRefGoogle Scholar
  5. 6.
    Janson S., Wolff T., Schatten classes and commutators of singular integrals, Ark. Mat. 20 (1982), 301–310.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 7.
    Luecking D., A characterization of certain classes of Hankel operators on the Bergman space of the unit disk, preprint, 1990.Google Scholar
  7. 8.
    Lin P., Rochberg R., The essential norm of Hankel operators on the Bergman space, Integral Equat. Oper. Theory (to appear).Google Scholar

References

  1. 1.
    Böttcher A., Silbermann B., Invertibility and Asymptotics of Toeplitz Matrice, Mathematische Forschung 17, Akademie-Verlag, Berlin, 1983.zbMATHGoogle Scholar
  2. 2.
    Böttcher A., Silbermann B., The finite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols, Math. Nachr. 110 (1983), 279–291.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Gohberg I. C., Feld'man I. A., Convolution Equations and Projection Methods for their Solution, Nauka, Moscow, 1971 (Russian); English translation: Amer. Math. Soc., Providence, R. I., 1974Google Scholar
  4. 4.
    Roch S., Silbermann B., Das Reduktionsverfahrens für Potenzen von Toeplitzoperatoren mit unstetigem Symbol, Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 24 (1982), 289–294.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Roch S., Silbermann B., Toeplitz-like operators, quasicommutator ideals, numerical analysis, Part I, Math. Nachr. 120 (1985), 141–173; Part II, Math. Nachr. 134 (1987), 381–391.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Silbermann B., Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren, Math. Nachr. 104 (1981), 137–146.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7ad.
    Verbitskiį I. È., On the reduction method for steps of Toeplitz matrices, Mat. Issl. 47 (1978), 3–11. (Russian)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Jaak Peetre
    • 1
  1. 1.Matematiska institutionenStockholmSweden

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