A description of Hankel operators of class $\mathfrak S_p$ for $p>0$, an investigation of the rate of rational approximation, and other applications

V. V. Peller

Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 465-494

Résumé

The main result is the following description of Hankel operators in the Schatten-von Neumann class $\mathfrak{S}_p$ when $0<p<1$: $$ \Gamma_\varphi\in\mathfrak S_p\Leftrightarrow\varphi\in B_p^{1/p}, $$ where $\Gamma_\varphi$ is the Hankel operator with symbol $\varphi$, and $B_p^{1/p}$ is the Besov class. This result extends results obtained earlier for $1\leqslant p<+\infty$ by the author to the case $ 0<p<1$. Also described are the Hankel operators in the Schatten–Lorentz classes $\mathfrak S_{pq}$, $0<p<+\infty$, $ 0<q\leqslant\infty$. Precise descriptions of classes of functions defined in terms of rational approximation in the bounded mean oscillation norm are given as an application, along with a complete investigation of the case where the decrease is of power order, and some precise results on rational approximation in the $L^\infty$-norm. Certain other applications are also considered. Bibliography: 57 titles.

Bibliographie

[1] Adamyan V. M., Arov D. Z., Krein M. G., “O beskonechnykh gankelevykh matritsakh i obobschennykh zadachakh Karateodori–Feiera i F. Rissa”, Funkts. analiz, 2:2 (1968), 1–19 | MR | Zbl

[2] Adamyan V. M., Arov D. Z., Krein M. G., “Analiticheskie svoistva par Shmidta gankeleva operatora i obobschennaya zadacha Shura–Takagi”, Matem. sb., 86 (128) (1971), 34–75

[3] Berg I., Lefstrem I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[4] Brudnyi Yu. A., “Prostranstva, opredelyaemye s pomoschyu lokalnykh priblizhenii”, Trudy Mosk. Matem. ob-va, 24 (1971), 64–132 | MR

[5] Brudnyi Yu. A., “Ratsionalnaya approksimatsiya i teoremy vlozheniya”, DAN SSSR, 247 (1979), 269–272 | MR | Zbl

[6] Vinogradov S. A., “Svoistva multiplikatorov integralov tipa Koshi–Stiltesa i nekotorye zadachi faktorizatsii analiticheskikh funktsii”, Trudy VII Zimnei Shkoly, Drogobych, 1974, Teor. funkts. i funkts. anal., M., 1976, 5–39 | MR | Zbl

[7] Gonchar A. A., “Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii”, Trudy mezhdunarodnogo kongressa matematikov, (Moskva, 1966), Mir, M., 1968, 329–346 | MR

[8] Gonchar A. A., Grigoryan L. D., “Ob otsenkakh normy golomorfnoi sostavlyayuschei meromorfnoi funktsii”, Matem. sb., 99 (141) (1976), 634–638 | Zbl

[9] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[10] Grigoryan L. D., “Otsenki normy golomorfnykh sostavlyayuschikh meromorfnykh funktsii v oblastyakh”, Matem. sb., 100 (142) (1976), 156–164 | Zbl

[11] Dolzhenko E. P., “Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii”, Matem. sb., 56 (98) (1962), 403–432 | Zbl

[12] Dynkin E. M., “Konstruktivnaya kharakteristika klassov S. L. Soboleva i O. V. Besova”, Trudy MIAN, 155, 1981, 41–76 | MR | Zbl

[13] Dynkin E. M., “O klassakh $B_{p^s}$ pri $01$”, DAN SSSR, 1983

[14] Ibragimov I. A., Rozanov Yu. A., Gaussovskie sluchainye protsessy, Nauka, M., 1970 | MR

[15] Ibragimov I. A., Solee V. N., “Nekotorye analiticheskie problemy, voznikayuschie v teorii statsionarnykh sluchainykh protsessov”, Zapiski nauchn. semin. LOMI, 81, 1978, 70–72 | MR

[16] Kislyakov S. V., “O proektorakh na mnozhestvo gankelevykh matrits”, Zapiski nauchn. semin. LOMI, 126, 1983, 109–116 | MR | Zbl

[17] Karadzhov G. E., “O primenenii teorii interpolyatsionnykh prostranstv k otsenkam singulyarnykh chisel integralnykh operatorov”, Problemy matem. analiza, 4, izd-vo LGU, 1973, 37–45

[18] Mergelyan S. N., Nekotorye voprosy konstruktivnoi teorii funktsii, Trudy MIAN, 37, 1951 | MR | Zbl

[19] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980 | MR

[20] Nikolskii N. K., Operatory Gankelya i Teplitsa, Preprinty LOMI, R-1-92, 2-2-82, Leningrad, 1982

[21] Pekarskii A. A., “Otsenki proizvodnoi integrala tipa Koshi s meromorfnoi plotnostyu i ikh prilozheniya”, Matem. zametki, 31:3 (1982), 389–402 | MR | Zbl

[22] Peller V. V., “Gladkie operatory Gankelya i ikh prilozheniya (idealy $\mathfrak{S}_p$, klassy Besova, sluchainye protsessy)”, DAN SSSR, 252:1 (1980), 43–48 | MR | Zbl

[23] Peller V. V., “Operatory Gankelya klassa $\mathfrak{S}_p$ i ikh prilozheniya (ratsionalnaya approksimatsiya, gaussovskie protsessy, problemy mazhoratsii operatorov)”, Matem. sb., 113 (155) (1980), 538–581 | MR | Zbl

[24] Peller V. V., “Ratsionalnaya approksimatsiya i gladkost funktsii”, Zapiski nauchn. semin. LOMI, 107, 1982, 150–159 | MR | Zbl

[25] Peller V. V., Khruschev S. V., “Operatory Gankelya, nailuchshie priblizheniya i statsionarnye gaussovskie protsessy”, UMN, 37:1 (1982), 53–124 | MR | Zbl

[26] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, GITTL, M., L., 1950

[27] Rotfeld S. Yu., “Zamechaniya o singulyarnykh chislakh summy vpolne nepreryvnykh operatorov”, Funkts. analiz, 1:3 (1967), 95–96 | MR

[28] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[29] Stoilov P., Vinogradov S. A., “Otsenki normy golomorfnykh sostavlyayuschikh meromorfnykh funktsii v edinichnom kruge”, Dokl. Bolg. akad. nauk (Comptes rend, de l'Academie bulgare des Sciences), 34:7 (1981), 931–934 | MR | Zbl

[30] Aleksandrov A. B., “Essays on non locally convex Hardy classes”, Lect. Notes in Math., 864, 1981, 1–89 | DOI | MR | Zbl

[31] Carleson L., Jacobs S., “Best uniform approximation by analytic functions”, Arkiv für Mathematik, 10:2 (1972), 219–229 | DOI | MR | Zbl

[32] Coifman R. R., Rochberg R., “Representation theorems for holomorphie and harmonic functions in $L^p$”, Asterisque, 77, 1980, 11–66 | MR | Zbl

[33] Duren P. L., Theory of $H^p$ spaces, Acad. Press., N.-Y., London, 1970 | MR

[34] Fefferman C., “Characterizations of bounded mean oscillations”, Bull. Amer. Math. Soc., 77 (1971), 587–588 | DOI | MR | Zbl

[35] Hartman P., “On completely contin uous Hankel matrices”, Proc. Amer. Math. Soc., 9:6 (1958), 862–866 | DOI | MR

[36] Helson H., Sarason D., “Past and Future”, Math. Scand., 21:1 (1967), 5–16 | MR

[37] Horowitz C., “Factorization theorems for functions in the Bergman spaces”, Duke-Math. J., 44:1 (1977), 201–213 | DOI | MR | Zbl

[38] Kahane J.-P., “Best approximations in $L^1(\mathbf{T})$”, Bull. Amer. Math. Soc., 80:5 (1974), 788–804 | DOI | MR | Zbl

[39] McCarthy, “$c_p$”, Israel Journ. Math., 5 (1967), 248–271 | MR

[40] Nehari Z., “On bounded bilinear forms”, Ann. of Math., 65:1 (1957), 153–162 | DOI | MR | Zbl

[41] Oswald P., On Besov–Hardy–Sobolev spaces of analytic functions in the unit disc, Preprint, 1981 | MR

[42] Peetre J., “Remarques sur les espaces de Besov”, Le cas S. R. Acad. Sc. Paris, 277:19 (1973), 947–949 | MR | Zbl

[43] Peetre J., New thoughts on Besov spaces, Duke. Univ. Math. Series 1. Math. Dep., Duke Univ., Durham, 1976 | MR | Zbl

[44] Peetre J., Svensson, On the generalized Hardy's inequality of McGehee, Pigno and Smith and the problem of interpolation between BMO and a Besov space, Univ. of Lund, Preprint, 10, Sweden, 1981 | Zbl

[45] Peller V. V., Nuclearity of Hankel operators. Adendum: Hankel operators of the class $\mathfrak{S}_p$ and projecting $\mathfrak{S}_p$ into the Hankel operators, LOMI Preprint, E-l-79, Leningrad, 1979 | MR

[46] Peller V. V., “Vectorial Hankel operators, commutators and related operators of the Schatten–von Neuman class $\mathfrak{S}_p$”, Integral Equat. and Oper. Theory 5, 1982, no. 2, 244–272 | DOI | MR | Zbl

[47] Peller V. V., Hankel operators of the Schatten–von Neumann class $\mathfrak{S}_p$, $0

1$, LOMI Preprint E-6-82, Leningrad, 1982

[48] Power S. C., Hankel operators on Hilbert space. Pitman Advanced Publishing Program 64, Boston, London, Melgourne, 1982 | MR

[49] Rochberg R., Trace ideals criteria for Hankel operators and commutators, Preprint, 1981 | MR

[50] Sarason D., “An addendum to “Past and Future””, Math. Scand., 30:1 (1972), 62–64 | MR | Zbl

[51] Sarason D., “Functions of vanishing mean oscillations”, Trans. Amer. Math. Soc., 207 (1973), 286–299 | MR

[52] Sarason D., Function theory on the unit circle, Notes for lectures at a conference at Virginia Polytechnic and State Univ., Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978 | MR

[53] Semmes S., Trace ideal criteria for Hankel operators, $0

1$, Preprint, 1982

[54] Shapiro H. S., Extramal problems for polynomials and power series, Dissertation, I, T., M., 1952

[55] Simon B., “Trace ideals and their applications”, L. M. S. Lect. Note Series No 35, Cambridge Univ. Pree, Cambridge, 1979 | MR | Zbl

[56] Simon B., “Pointwise domination of matrices and comparison of $\mathfrak{S}_p$ norms”, Pacif J. Math., 97:2 (1981), 471–475 | MR | Zbl

[57] Wolff T., “A note on interpolation spaces”, Lect. Notes in Math., 908 (1982), 199–204 | DOI | MR | Zbl