Ha-Plitz Operators: A Survey of Some Recent Results

Part of the NATO ASI series book series (ASIC, volume 153)


Originally, Hankel and Toeplitz operators are defined as operators acting on ℓ2 and having matrices with entries depending only on the sum or, respectively, difference of indices: \( \Gamma = {\left\{ {{\gamma_{n + k}}} \right\}_{n,k \geqslant 0}} \), \( T = \left\{ {{t_{n - k}}} \right\}{}_{n,k \geqslant 0} \). Many close relations of such operators (matrices) to various problems of algebra, analysis, differential equations were discovered as early as in the last century. But the spectral theories of Hankel and Toeplitz operators start their development only in the late 50th. Now they are joined within the spectral theory of Hankel and Toeplitz operators.


Toeplitz Operator Interpolation Problem Blaschke Product Riesz Basis Hankel Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Adamyan, V.M.; Arov D.Z. and Krein, M.G. Аналитические свойства пар Шмидта ганкелева оператора и обобщенная задача Щура-Такаги. — Матем.сборник, 1971, 85(128), No 1(9), рр.33–73.Google Scholar
  2. [2]
    Adamyan, V.M.; Arov, D.Z. and Krein, M.G. Бесконечные блочно-ганкелевы матрицы и связанные с ними проблемы продолжения. — Известия АН Армянской ССР, Математика, 1971, 2–3, pp.87–112.Google Scholar
  3. [3]
    Axler, S. Factorization of L∞ functions. — Ann. Math., 1977, 10, pp.567–572.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Ball, J.A. and Helton, J.W. A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation theory. — J.Operator Theory, 1983, 9, рр.107–142.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Bram, J. Subnormal operators. — Duke Math.J., 1955, 22, 1, pp.75–94.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Clark, D.N. On the point spectrum of a Toeplitz operator. — Trans.Amer.Math.Soc., 1967, 126, 2, pp.251–266.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Clark, D.N. On Toeplitz operators with loops. — J. Operator Theory, 1980, 4, pp.37–54.zbMATHGoogle Scholar
  8. [8]
    Clark, D.N. On Toeplitz operators with loops. II. — J. Operator Theory, 1982, 7, pp.109–123.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Douglas, R.G. Banach algebra techniques in operator theory. Academic Press, N.Y., 1972.zbMATHGoogle Scholar
  10. [10]
    Douglas R.G. Banach algebra techniques in the theory of Toeplitz operators. — CBMS series, 15, Amer.Math.Soc., Providence, 1973.zbMATHGoogle Scholar
  11. [11]
    Helton, J.W. Bon-Euclidean functional analysis and electronics. — Bull.Amer.Math.Soc., 1982, 7, 1, pp.1–64.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Hruščev, S.V.; Nikol’skii, N.K. and Pavlov, B.S. Unconditional bases of exponentials and of reproducing kernels. — Lect. Notes Math., 864, Springer -Verlag, Berlin — N.Y., 1981, pp.214–335.Google Scholar
  13. [13]
    Hruščev, S.V. and Peller, V.V. Moduli of Hankel operators, past and future. — Lect. Notes Math., 1043, Springer-Verlag, Berlin — N.Y., 1984, pp.92–97.Google Scholar
  14. [14]
    Iohvidov, I.S. Об одной лемме К.Фана, обобщающей цринцип неподвижной точки А.Н.Тихонова. — Докл.АН СССР, I964, I59, 3, pp.50I–504.Google Scholar
  15. [15]
    Ку Fan. Generalization of Tychonoff fixed point theorem. — Math.Annalen, 1961, 142, pp.305–31.zbMATHCrossRefGoogle Scholar
  16. [16]
    Linear and Complex Analysis Problem Book. Lect. Notes Math., 1043, Springer-Verlag, Berlin-N.Y. 1984.Google Scholar
  17. [17]
    Nikol’skii, N.K. Лекции об операторе сдвига. Изд. “Наука”, Москва, 1980. (English translation including [18] as an Appendix: Treatise on the shift operator. Springer-Verlag, Berlin — N.Y., 1984–1985).Google Scholar
  18. [18]
    Nikol’skii, N.K. Операторы Ганкеля и Теплица. Спектральная теория. Алгебраический подход. Последние достижения. — Препринты ЛОМИ (Ленинград) P-I-82, Р-2–82, Р-5–82, рр.I–I81.Google Scholar
  19. [19]
    Nikol’skii, N.К. Наброски к вычислению кратности спектра ортогональных сумм. — Записки научи.семинаров ЛОМИ (Ленинград), I983, I26, pp.I50–I58.Google Scholar
  20. [20]
    Nikol’skii, N.K. and Vasyunin, V.I. Control sub-spaces of minimal dimension and root vectors. — Integral Eq. and Operator Theory, 1983, 6, pp.274–311.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Page, L.B. Applications of Sz.-Nagy and Poiaş lifting theorem. — Indiana Univ.Math.J., 1970, 20, рр.135–145.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Peetre, J. Hankel operators, rational approximation and allied questions of analysis. — Canadian Math.Soc.Confer.Proc., 1983, 3, 287–332.MathSciNetGoogle Scholar
  23. [23]
    Peller, V.V. Операторы Ганкеяя класса (math) их приложения (рациональная апцроксимация, гауссовские процессы, проблема мажорации операторов) — Матем.сборник, I980, II3, 4, рр.538–58I.MathSciNetGoogle Scholar
  24. [24]
    Peller, V.V. and Hruščev, S.V. Операторы Ганке-ля, наилучшие приближения и стационарные гауссовские процессы. — Успехи матем.наук, I982,37, I, pp.53–I24.Google Scholar
  25. [25]
    Power, S.С. Hankel operators on Hilbert space. Research Notes in Math., 64, Pitman, Boston London — Melbourne, 1982.zbMATHGoogle Scholar
  26. [26]
    Power, S.C. Quasinilpotent Hankel operators. — Lect. Notes Math., 1043, Springer-Verlag, Berlin — N.Y., pp.259–261.Google Scholar
  27. [27]
    Redheffer, R.M. Completeness of sets of complex exponentials. — Adv.Math., 1977, 24, pp.1–62.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Sarason, D.E. Function theory on the unit circle — Notes for lectures at a conf. at Virginia Polytechnic Inst. and State Univ., 1978, preprint.zbMATHGoogle Scholar
  29. [29]
    Sarason, D.E. Algebras of functions on the unit circle. — Bull.Amer.Math.Soc., 1973, 79, pp.286-289.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Sarason, D.E. Weak-star generators of H∞ — Pacif.J.Math., 1966, 17, N 3, pp.519–528.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Sarason, D.E. Generalized interpolation in H∞. — Trans.Amer.Math.Soc., 1967, 127, N 2, рр.179–203.MathSciNetzbMATHGoogle Scholar
  32. [32]
    Sedleckii, A.M. Биортогональные разложения функций в ряды экспонент на интервалах вещественной оси. — Успехи Матем.Наук, 1982, 37, 5, pp.51–95.Google Scholar
  33. [33]
    Solomyak, B.M. О кратности спектра аналитических операторов Теплица. — Докл.АН СССР, 1985.Google Scholar
  34. [34]
    Spitkovskii, I.M. О множителях, не влияющих на факторизуемость. — Матем.заметки, 1980, 27, No 2, pp.291–299.MathSciNetGoogle Scholar
  35. [35]
    Treil, S.R. Векторный вариант теоремы Адамяна-Арова-Крейна. — Функциональный анализ и его приложения, 1985.Google Scholar
  36. [36]
    Treil, S.R. Модули операторов Ганкеля и задача В.В.Пеллера-С.В.Хрущева. — Докл.АН СССР, 1985.Google Scholar
  37. [37]
    Volberg, A.L. How to break through a prescribed contour. — To appear.Google Scholar
  38. [38]
    Wolff, Т.Н. Two algebras of bounded functions. — Duke Math.J., 1982, 49, 2, pp.321–328.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  1. 1.Leningrad BranchSteklov Math InstituteLeningradUSSR

Personalised recommendations