Geodesic
Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)
Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 443-479 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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; 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. 
@article{SM_1982_41_4_a2,
     author = {V. V. Peller},
     title = {Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, {Gaussian} processes, the problem of majorizing operators)},
     journal = {Sbornik. Mathematics},
     pages = {443--479},
     year = {1982},
     volume = {41},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1982_41_4_a2/}
}
TY  - JOUR
AU  - V. V. Peller
TI  - Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)
JO  - Sbornik. Mathematics
PY  - 1982
SP  - 443
EP  - 479
VL  - 41
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1982_41_4_a2/
LA  - en
ID  - SM_1982_41_4_a2
ER  - 
%0 Journal Article
%A V. V. Peller
%T Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)
%J Sbornik. Mathematics
%D 1982
%P 443-479
%V 41
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1982_41_4_a2/
%G en
%F SM_1982_41_4_a2
V. V. Peller. Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators). Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 443-479. http://geodesic.mathdoc.fr/item/SM_1982_41_4_a2/

[1] N. K. Nikolskii, Lektsii ob operatore sdviga, izd-vo “Nauka”, Moskva, 1980 | MR

[2] I. A. Ibragimov, Yu. A. Rozanov, Gaussovskie sluchainye protsessy, izd-vo “Nauka”, Moskva, 1970 | MR

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

[4] V. M. Adamyan, D. Z. Arov, M. G. Krein, “Beskonechnye gankelevy matritsy i obobschennye zadachi Karateodori–Feiera i I. Shura”, Funkts. analiz, 2:4 (1968), 1–17 | Zbl

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

[6] D. Sarason, Function theory on the unit disc, Lect. notes Univ. Virginia, 1978

[7] H. Widom, “Hankel matrices”, Trans. Amer. Math. Soc., 121:1 (1966), 1–35 | DOI | MR | Zbl

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

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

[10] J. M. Anderson, K. F. Barth, D. A. Brannan, “Research problems in complex analysis”, Bull. London Math. Soc., 9, part 2:26 (1977), 129–162 | DOI | MR | Zbl

[11] J. S. Howland, “Trace class Hankel operators”, Quart J. Math., Oxford, (2)22:85 (1971), 147–159 | DOI | MR

[12] I. A. Ibragimov, V. N. Solev, “Nekotorye analiticheskie problemy, voznikayuschie v teorii statsionarnykh sluchainykh protsessov”, 99 nereshennykh zadach lineinogo i kompleksnogo analiza, Zapiski nauchnykh seminarov LOMI, 81, 1978, 70–72

[13] B. Simon, Trace ideals and their applications, Lect. Notes London Math. Soc., Cambridge Univ. Press, 1979 | MR | Zbl

[14] V. V. Peller, “Gladkie gankelevy operatory i ikh prilozheniya (idealy $\mathfrak G_p$, klassy Besova, sluchainye protsessy)”, DAN SSSR, 252:1 (1980), 43–48 | MR | Zbl

[15] V. V. Peller, Nuclearity of Hankel operators, Preprints E-I-79, LOMI, Leningrad, 1979 | MR | Zbl

[16] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, izd-vo “Nauka”, Moskva, 1965

[17] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, izd-vo “Nauka”, Moskva, 1977 | MR

[18] I. Stein, Singulyarnye integraly i differentsialnye svoistva funktsii, izd-vo “Mir”, Moskva, 1973 | MR

[19] H. Triebel, Interpolation theory, Function spaces. Differential operators, VEB, Deutsch. Verl. Wiss., Berlin, 1978 | MR | Zbl

[20] V. P. Zakharyuta, V. I. Yudovich, “Obschii vid lineinogo funktsionala v $H^p$”, Uspekhi matem. nauk, XIX:2(116) (1964), 139–142

[21] P. L. Duren, B. W. Ronberg, A. L. Shields, “Linear functionals on $H^p$-spaces with $0

1$”, J. reine und angew. Math., 238 (1969), 32–60 | MR | Zbl

[22] A. Zigmund, Trigonometricheskie ryady, t. I, II, izd-vo “Mir”, Moskva, 1965 | MR

[23] J. Arazy, J. Lindenstrauss, “Some linear topological properties of the spaces $C_p$ of operators on Hilbert space”, Compositio Math., 30:1 (1975), 81–111 | MR | Zbl

[24] S. Kwapień, A. Pelczyński, “Some linear topological properties of the Hardy spaces $H_p$”, Compositio Math., 33:3 (1976), 261–288 | MR | Zbl

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

[26] D. Sarason, “Generalized interpolation in $H^\infty$”, Trans. Amer. Math. Soc., 127:2 (1967), 179–203 | DOI | MR | Zbl

[27] L. Kronecker, “Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen”, Manatsber. königl. Preussischen Akad. Wiss. Berlin, 1881, 535–600 | Zbl

[28] C. Fefferman, E. M. Stein, “$H^p$ spaces of several variables”, Acta Math., 129 (1972), 137–193 | DOI | MR | Zbl

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

[30] A. A. Gonchar, “Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii”, Trudy mezhdun. kongressa matematikov (Moskva, 1966), izd-vo “Mir”, Moskva, 1968, 329–356

[31] Yu. A. Brudnyi, “Ratsionalnaya approksimatsiya i teoremy vlozheniya”, DAN SSSR, 247:2 (1979), 269–272 | MR | Zbl

[32] S. G. Krein, Yu. I. Petunii, E. M. Semenov, Interpolyatsiya lineinykh operatorov, izd-vo “Nauka”, Moskva, 1978 | MR

[33] N. V. Miroshin, “Nekotorye svoistva interpolyatsionnykh prostranstv vpolne nepreryvnykh operatorov”, Matem. zametki, 17:2 (1975), 293–300 | MR | Zbl

[34] D. Sarason, “Functions of vanishing mean oscillation”, Trans. Amer. Math. Soc., 207 (1975), 391–405 | DOI | MR | Zbl

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

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

[37] I. A. Ibragimov, V. N. Solev, “Ob odnom uslovii regulyarnosti gaussovskoi statsionarnoi posledovatelnosti”, Zapiski nauchnykh seminarov LOMI, 12, 1969, 113–125 | MR | Zbl

[38] K. Gofman, Banakhovy prostranstva analiticheskikh funktsii, IL, Moskva, 1963

[39] V. N. Solev, “Absolyutno regulyarnye traektorii v gilbertovom prostranstve”, Zapiski nauchnykh seminarov LOMI, 22, 1971, 139–160 | MR | Zbl

[40] V. N. Solev, “Ob odnom uslovii lineinoi regulyarnosti statsionarnoi vektornoi posledovatelnosti”, Zapiski nauchnykh seminarov LOMI, 12, 1969, 126–145 | MR | Zbl

[41] P. Khalmosh, Gilbertovo prostranstvo v zadachakh, izd-vo “Mir”, Moskva, 1970 | MR

[42] R. G. Douglas, Banach algebra techniques in operator theory, New York, London, 1972 | MR

[43] L. D. Pitt, “A compactness condition for linear operators on function spaces”, J. operator theory, 1:1 (1979), 49–54 | MR | Zbl

[44] R. P. Boas, “Majoraut problems for Fourier series”, J. Analyse Math., 10 (1962), 253–271 | DOI | MR

[45] G. F. Bachelis, “On the upper and lower majoraut properties in $L^p(G)$”, Quart. J. Math., Oxford, (2)24 (1973), 119–128 | DOI | MR

[46] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, Moskva, 1958

[47] E. M. Dynkin, “Konstruktivnaya kharakteristika klassov S. L. Soboleva i O. V. Besova”, Trudy matem. in-ta im. V. A. Steklova, CLV, 1981, 41–76 | MR | Zbl