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Mat. Sb. (N.S.), 1980, Volume 113(155), Number 4(12), Pages 538–581 (Mi msb2817)  

This article is cited in 85 scientific papers (total in 85 papers)

Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)

V. V. Peller

Abstract: A criterion is given for a Hankel operator $H_\varphi\colon H^2\to H^2_-$ ($H_\varphi f=(I-\mathbf P)\varphi f$, where $\mathbf P$ is the orthogonal projection of $L^2$ onto $H^2$) to belong to the Schatten–von Neumann class $\mathfrak S_p$ in terms of its symbol $\varphi$. Various applications are considered: a precise description is obtained for classes of functions definable in terms of rational approximation in the $BMO$ (bounded mean oscillation) norm; it is proved that the averaging projection onto the set of Hankel operators is bounded in the norm of $\mathfrak S_p$, $1<p<+\infty$; a counterexample is given to a conjecture of Simon on the majorization property in $\mathfrak S_p$; a problem of Ibragimov and Solev on stationary Gaussian processes is solved; and a criterion is obtained for functions of an operator in the Sz.-Nagy–Foias model to belong to the class $\mathfrak S_p$.
Bibliography: 47 titles.

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Mathematics of the USSR-Sbornik, 1982, 41:4, 443–479

Bibliographic databases:

UDC: 517.5
MSC: Primary 30D55, 46E35, 47B10, 47B35; Secondary 30E05, 41A20, 41A25, 47D25, 60G10, 60G15
Received: 25.03.1980

Citation: V. V. Peller, “Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)”, Mat. Sb. (N.S.), 113(155):4(12) (1980), 538–581; Math. USSR-Sb., 41:4 (1982), 443–479

Citation in format AMSBIB
\by V.~V.~Peller
\paper Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)
\jour Mat. Sb. (N.S.)
\yr 1980
\vol 113(155)
\issue 4(12)
\pages 538--581
\jour Math. USSR-Sb.
\yr 1982
\vol 41
\issue 4
\pages 443--479

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    This publication is cited in the following articles:
    1. V. V. Peller, S. V. Khrushchev, “Hankel operators, best approximations, and stationary Gaussian processes”, Russian Math. Surveys, 37:1 (1982), 61–144  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Janson S., Wolff T., “Schatten Classes and Commutators of Singular Integral-Operators”, Ark. Mat., 20:2 (1982), 301–310  crossref  mathscinet  zmath  isi
    3. A. V. Bukhvalov, “Application of methods of the theory of order-bounded operators to the theory of operators in $L^p$-spaces”, Russian Math. Surveys, 38:6 (1983), 43–98  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. V. V. Peller, “A description of Hankel operators of class $\mathfrak S_p$ for $p>0$, an investigation of the rate of rational approximation, and other applications”, Math. USSR-Sb., 50:2 (1985), 465–494  mathnet  crossref  mathscinet  zmath
    5. Clark D., “Hankel-Operators on Hilbert-Space - Power,Sc”, Bull. Amer. Math. Soc., 9:1 (1983), 98–102  crossref  mathscinet  isi
    6. Pekarskii A., “Rational Approximation of the Class Hp, O Greater-Than-P-Greater-Than-Infinity”, Dokl. Akad. Nauk Belarusi, 27:1 (1983), 9–12  mathscinet  isi
    7. Semmes S., “Another Characterization of Hp, O-Less-Than-P-Less-Than-Infinity, with an Application to Interpolation”, Lect. Notes Math., 992 (1983), 212–226  crossref  mathscinet  isi
    8. Peller V., “Continuity Properties of the Averaging Projection Onto the Set of Hankel-Matrices”, J. Funct. Anal., 53:1 (1983), 74–83  crossref  mathscinet  zmath  isi
    9. A. A. Pekarskii, “Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation”, Math. USSR-Sb., 52:2 (1985), 557–574  mathnet  crossref  mathscinet  zmath
    10. L. D. Pustyl'nikov, “Toeplitz and Hankel matrices and their applications”, Russian Math. Surveys, 39:4 (1984), 63–98  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. Michele Pavon, “Canonical correlations of past inputs and future outputs for linear stochastic systems”, Systems & Control Letters, 4:4 (1984), 209  crossref
    12. Pekarskii A., “Direct and Inverse-Theorems of Rational Approximation of the Hardy Class”, Dokl. Akad. Nauk Belarusi, 28:2 (1984), 111–114  mathscinet  isi
    13. Peller V., “Metric Properties of an Averaging Projector Onto the Sets of Hankel-Matrices”, 278, no. 2, 1984, 275–281  mathscinet  zmath  isi
    14. Peetre J., Svensson E., “On the Generalized Hardy Inequality of Mcgehee, Pigno and Smith and the Problem of Interpolation Between Bmo and a Besov Space”, Math. Scand., 54:2 (1984), 221–241  mathscinet  zmath  isi
    15. Janson S., Nilsson P., Peetre J., “Notes on Wolff Note on Interpolation Spaces”, Proc. London Math. Soc., 48:MAR (1984), 283–299  crossref  mathscinet  zmath  isi
    16. Janson S., Peetre J., “Higher-Order Commutators of Singular Integral-Operators”, 1070, 1984, 125–142  mathscinet  zmath  isi
    17. Ibragimov I., Solev V., “Some Analytical Problems in the Theory of Stationary Stochastic-Processes”, 1043, 1984, 87–91  isi
    18. Hruscev S., Peller V., “Moduli of Hankel-Operators, Past and Future”, 1043, 1984, 92–97  mathscinet  isi
    19. A. A. Pekarskii, “Classes of analytic functions determined by best rational approximations in $H_p$”, Math. USSR-Sb., 55:1 (1986), 1–18  mathnet  crossref  mathscinet  zmath
    20. V. V. Peller, “Hankel operators in the perturbation theory of unitary and self-adjoint operators”, Funct. Anal. Appl., 19:2 (1985), 111–123  mathnet  crossref  mathscinet  zmath  isi
    21. Finbarr Holland, “Report on the Dublin matrix theory conference, March 1984”, Linear Algebra and its Applications, 68 (1985), 263  crossref
    22. Power S., “Commutators with the Triangular Projection and Hankel Forms on Nest-Algebras”, J. Lond. Math. Soc.-Second Ser., 32:2 (1985), 272–282  crossref  mathscinet  zmath  isi
    23. O. G. Parfenov, “Estimates of the singular numbers of the Carleson imbedding operator”, Math. USSR-Sb., 59:2 (1988), 497–514  mathnet  crossref  mathscinet  zmath
    24. Richard Rochberg, Stephen Semmes, “A decomposition theorem for BMO and applications”, Journal of Functional Analysis, 67:2 (1986), 228  crossref
    25. Timotin D., “A Note on Cp Estimates for Certain Kernels”, Integr. Equ. Oper. Theory, 9:2 (1986), 295–304  crossref  mathscinet  zmath  isi
    26. A. A. Pekarskii, “Tchebycheff rational approximation in the disk, on the circle, and on a closed interval”, Math. USSR-Sb., 61:1 (1988), 87–102  mathnet  crossref  mathscinet  zmath
    27. A. A. Gonchar, E. A. Rakhmanov, “Equilibrium distributions and degree of rational approximation of analytic functions”, Math. USSR-Sb., 62:2 (1989), 305–348  mathnet  crossref  mathscinet  zmath  isi
    28. Volberg A. Ivanov O., “Belonging of the Product of 2 Hankel-Operators to the Schatten-Vonneumann Class”, no. 4, 1987, 3–6  mathscinet  zmath  isi
    29. Janson S., Peetre J., “A New Generalization of Hankel-Operators - (the Case of Higher Weights)”, Math. Nachr., 132 (1987), 313–328  crossref  mathscinet  zmath  isi
    30. Janson S., Peetre J., “Paracommutators - Boundedness and Schatten-Vonneumann Properties”, Trans. Am. Math. Soc., 305:2 (1988), 467–504  crossref  mathscinet  zmath  isi
    31. Petrushev P., “Direct and Converse Theorems for Spline and Rational Approximation and Besov-Spaces”, Lect. Notes Math., 1302 (1988), 363–377  crossref  mathscinet  zmath  isi
    32. Peller V., “Smoothness of Schmidt Functions of Smooth Hankel-Operators”, Lect. Notes Math., 1302 (1988), 337–346  crossref  mathscinet  zmath  isi
    33. Timotin D., “Cp-Estimates for Certain Kernels on Local-Fields”, Studia Math., 88:1 (1988), 43–50  mathscinet  zmath  isi
    34. Richard Rochberg, Stephen Semmes, “Nearly weakly orthonormal sequences, singular value estimates, and Calderon-Zygmund operators”, Journal of Functional Analysis, 86:2 (1989), 237  crossref
    35. Peetre J. Karlsson J., “Rational Approximation-Analysis of the Work of Pekarskii”, Rocky Mt. J. Math., 19:1 (1989), 313–333  crossref  mathscinet  zmath  isi
    36. Janson S., Peetre J., Wallsten R., “A New Look on Hankel Forms Over Fock Space”, Studia Math., 95:1 (1989), 33–41  mathscinet  zmath  isi
    37. C. K. Chui, X. Li, J. D. Ward, “On the convergence rate ofs-numbers of compact Hankel operators”, Circuits Syst Signal Process, 11:2 (1992), 353  crossref  mathscinet  zmath  isi
    38. Netrusov Y., “Interpolation (Real Method) of Spaces of Smooth Functions with Space of Bounded-Functions”, Dokl. Akad. Nauk, 325:6 (1992), 1120–1123  mathnet  mathscinet  zmath  isi
    39. V. A. Prokhorov, “Rational approximation of analytic functions”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 139–164  mathnet  crossref  mathscinet  zmath  isi
    40. A. Khatamov, “Inverse theorems in the theory of rational approximations of functions of several variables”, Math. Notes, 54:2 (1993), 858–866  mathnet  crossref  mathscinet  zmath  isi
    41. A. P. Petukhov, “Convergence of Fourier series for functions in the classes of Besov–Lizorkin–Triebel”, Math. Notes, 56:1 (1994), 694–698  mathnet  crossref  mathscinet  zmath  isi
    42. V. L. Kreptogorskii, “Interpolation in Lizorkin–Triebel and Besov spaces”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 315–326  mathnet  crossref  mathscinet  zmath  isi
    43. Qingtang Jiang, Lizhong Peng, “Toeplitz and Hankel type operators on an annulus”, Mathematika, 41:2 (1994), 266  crossref  isi
    44. Stepanov V., “On Singular Numbers of a Certain Class of Integral-Operators”, Dokl. Akad. Nauk, 336:4 (1994), 457–458  mathnet  mathscinet  zmath  isi
    45. Wysoczanski J., “A Characterization of Radial Herz-Schur Multipliers on Free-Products of Discrete-Groups”, J. Funct. Anal., 129:2 (1995), 268–292  crossref  mathscinet  zmath  isi
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    50. A HARCHARRAS, “Analyse de Fourier, multiplicateurs de Schur sur Sp et ensembles Λ(p)cb non commutatifs”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 326:7 (1998), 845  crossref
    51. È. S. Belinskii, “Interpolation and integral norms of hyperbolic polynomials”, Math. Notes, 66:1 (1999), 16–23  mathnet  crossref  crossref  mathscinet  zmath  isi
    52. Bonami A., Peloso M., Symesak F., “Powers of the Szego Kernel and Hankel Operators on Hardy Spaces”, Mich. Math. J., 46:2 (1999), 225–250  crossref  mathscinet  zmath  isi
    53. William Hornor, James E. Jamison, “Isometries of some Banach spaces of analytic functions”, Integr equ oper theory, 41:4 (2001), 410  crossref  mathscinet  zmath  isi
    54. Basor E., Ehrhardt T., “Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices”, Math. Nachr., 228 (2001), 5–45  crossref  mathscinet  zmath  isi
    55. M. T. Karaev, “Berezin Symbols and Schatten–von Neumann Classes”, Math. Notes, 72:2 (2002), 185–192  mathnet  crossref  crossref  mathscinet  zmath  isi
    56. A. A. Pekarskii, “New Proof of the Semmes Inequality for the Derivative of the Rational Function”, Math. Notes, 72:2 (2002), 230–236  mathnet  crossref  crossref  mathscinet  zmath  isi
    57. Ho M., “Operators on Spaces of Analytic Functions Belonging to l-(1,l-Infinity)”, J. Math. Anal. Appl., 268:2 (2002), 665–683  crossref  mathscinet  zmath  isi
    58. Cwikel M., Persson L., Rochberg R., Sparr G., “Jaak Peetre, the Man and His Work”, Function Spaces, Interpolation Theory and Related Topics, Proceedings, eds. Cwikel M., Englis M., Kufner A., Persson L., Spaar G., Walter de Gruyter & Co, 2002, 1–22  crossref  mathscinet  isi
    59. Ehrhardt T., “A New Algebraic Approach to the Szego-Widom Limit Theorem”, Acta Math. Hung., 99:3 (2003), 233–261  crossref  mathscinet  zmath  isi
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    65. Ho M., Wong M., “Analytic Spaces Defined by Symmetric Norming Functions”, Taiwan. J. Math., 10:1, SI (2006), 1–11  mathscinet  zmath  isi
    66. L. Baratchart, M. L. Yattselev, “Meromorphic approximants to complex Cauchy transforms with polar singularities”, Sb. Math., 200:9 (2009), 1261–1297  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    69. Opmeer M.R., “Decay of Hankel Singular Values of Analytic Control Systems”, Syst. Control Lett., 59:10 (2010), 635–638  crossref  mathscinet  zmath  isi
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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