Some estimates for the partial indices of measurable matrix-valued functions

I. M. Spitkovsky

Sbornik. Mathematics, Tome 39 (1981) no. 2, pp. 207-226

Résumé

Tests are given for nonnegativity, nonpositivity, and stability of partial indices of measurable bounded $n\times n$ matrix-valued functions defined on a contour $\Gamma$ along which the operator $S$ of singular integration is bounded in the spaces $L_p$, $1<p<\infty$. In particular, a sufficient condition is given for the coincidence of the partial indices of a matrix-valued function $G$ formulated in terms of the Hausdorff set of the matrices $G(t)$, $t\in \Gamma$. As auxiliary results, necessary and sufficient conditions are given for the operators of the form $T_G=\frac12(I-S)|\operatorname{Im}(I-S)$ to be Fredholm, or $n$- or $d$-normal in the case $G\in E^\pm_\infty+C$, and the behavior of the factorization is studied under the multiplication by such matrix-valued functions $G$ ($E^\pm_\infty$ are the Smirnov classes in the domains with boundary $\Gamma$ and $C$ is the class of functions continuous on $\Gamma$). In the case where $\Gamma$ is the unit circle, for the factorization in $L_2$ necessary and sufficient conditions are found for the nonnegativity (nonpositivity, and so on) of the partial indices. For a Lyapunov contour $\Gamma$ a sufficient condition (which is also necessary for $p=2$) is formulated for the vectorial boundary value problem of Riemann to be Fredholm in the spaces $L^n_p$ and $L^n_q$ ($q=p/(p-1)$). Bibliography: 38 titles.

Bibliographie

[1] N. I. Muskhelishvili, Singulyarnye integralnye uravneniya, izd-vo “Nauka”, Moskva, 1968 | MR

[2] N. P. Vekua, Sistemy singulyarnykh integralnykh uravnenii, izd-vo “Nauka”, Moskva, 1970 | MR

[3] I. B. Simonenko, “Kraevaya zadacha Rimana dlya $n$ par funktsii s izmerimymi koeffitsientami i ee primenenie k issledovaniyu singulyarnykh integralov v prostranstvakh $L_p$ s vesom”, Izv. AN SSSR, seriya matem., 28:2 (1964), 277–306 | MR | Zbl

[4] I. B. Simonenko, “Nekotorye obschie voprosy teorii kraevoi zadachi Rimana”, Izv. AN SSSR, seriya matem., 32 (1968), 1138–1146 | MR | Zbl

[5] I. I. Danilyuk, Neregulyarnye granichnye zadachi na ploskosti, izd-vo “Nauka”, Moskva, 1975 | MR

[6] B. V. Khvedelidze, “Metod integralov tipa Koshi v razryvnykh granichnykh zadachakh teorii golomorfnykh funktsii odnoi kompleksnoi peremennoi”, Sovremennye problemy matematiki, 7, 1975, 5–162

[7] I. Ts. Gokhberg, M. G. Krein, “Sistemy integralnykh uravnenii na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, Uspekhi matem. nauk, XIII:2(80) (1958), 3–72

[8] I. Ts. Gokhberg, M. G. Krein, “Ob ustoichivoi sisteme chastnykh indeksov zadachi Gilberta dlya neskolkikh neizvestnykh funktsii”, DAN SSSR, 119:5 (1958), 854–857 | MR | Zbl

[9] B. V. Boyarskii, “Ob ustoichivosti zadachi Gilberta dlya golomorfnogo vektora”, Soobsch. AN Gruz SSR, 21:4 (1958), 391–398 | Zbl

[10] I. M. Spitkovskii, “Ustoichivost chastnykh indeksov kraevoi zadachi Rimana so strogo nevyrozhdennoi matritsei”, DAN SSSR, 218:1 (1974), 46–49 | MR | Zbl

[11] I. M. Spitkovskii, “Zadacha faktorizatsii izmerimykh matrits-funktsii”, DAN SSSR, 227:3 (1976), 576–579 | MR | Zbl

[12] I. M. Spitkovskii, “O chastnykh indeksakh nepreryvnykh matrits-funktsii”, DAN SSSR, 229:5 (1976), 1059–1062 | MR | Zbl

[13] I. M. Spitkovskii, “O faktorizatsii matrits-funktsii, khausdorfovo mnozhestvo kotorykh raspolozheno vnutri ugla”, Soobsch. AN GruzSSR, 86:3 (1977), 561–564 | MR | Zbl

[14] V. P. Khavin, “Granichnye svoistva integrala tipa Koshi i garmonicheski sopryazhennykh funktsii v oblastyakh so spryamlyaemoi granitsei”, Matem. sb., 68(110) (1965), 499–517 | Zbl

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

[16] G. Ts. Tumarkin, S. Ya. Khavinson, “K opredeleniyu analiticheskikh funktsii klassa $E_p$ v mnogosvyaznykh oblastyakh”, Uspekhi matem. nauk, XIII:1(79) (1958), 201–206

[17] M. Lee, D. Sarason, “The spectra of some Toeplitz operators”, J. Math. Anal. Appl., 33:3 (1971), 529–543 | DOI | MR | Zbl

[18] G. V. Ambartsumyan, “Ob obraschenii nekotorykh teplitsevykh matrits”, Matem. issled., 8:3 (1973), 140–144 | MR | Zbl

[19] R. G. Douglas, “Toeplitz and Wiener–Hopf operator in $H^\infty+C$”, Bull. Amer. Math. Soc., 26 (1970), 117–120 | MR

[20] G. V. Ambartsumyan, “O metode reduktsii dlya odnogo klassa teplitsevykh matrits”, Matem. issled., 8:2 (1973), 161–169 | MR | Zbl

[21] A. M. Nikolaichuk, “Nekotorye otsenki dlya chastnykh indeksov kraevoi zadachi Rimana”, Ukr. matem. zh., 23:6 (1971), 793–798

[22] I. E. Verbitskii, N. Ya. Krupnik, “Tochnye konstanty v teoremakh K. I. Babenko i B. V. Khvedelidze ob ogranichennosti singulyarnogo operatora”, Soobsch. AN GruzSSR, 85:1 (1977), 21–24 | MR

[23] I. M. Glazman, Yu. I. Lyubich, Konechnomernyi lineinyi analiz v zadachakh, izd-vo “Nauka”, Moskva, 1969 | MR

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

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

[26] V. E. Lyantse, “Nekotorye svoistva idempotentnykh operatorov”, Teor. i prikl. mekhanika, 1 (1958), 16–22, Lvov

[27] A. I. Maltsev, Osnovy lineinoi algebry, izd-vo “Nauka”, Moskva, 1970

[28] N. Ya. Krupnik, V. I. Nyaga, “O singulyarnykh integralnykh operatorakh v sluchae negladkogo kontura”, Matem. issled., 10:1 (1975), 144–164 | MR

[29] N. Ya. Krupnik, “O singulyarnykh integralnykh operatorakh s matrichnymi koeffitsientami”, Spektralnye svoistva operatorov, izd-vo “Shtiintsa”, Kishinev, 1977, 93–100 | MR

[30] B.-S. Nad, Ch. Foyash, Garmonicheskii analiz operatorov v gilbertovom prostranstve, izd-vo “Mir”, Moskva, 1970 | MR

[31] M. G. Krein, I. M. Spitkovskii, “O faktorizatsii matrits-funktsii na edinichnoi okruzhnosti”, DAN SSSR, 234:2 (1977), 287–290 | MR | Zbl

[32] M. G. Krein, I. M. Spitkovskii, “O faktorizatsii $\alpha$-sektorialnykh matrits-funktsii na edinichnoi okruzhnosti”, Operatory v banakhovykh prostranstvakh, izd-vo “Shtiintsa”, Kishinev, 1978, 41–63 | MR

[33] B. L. Page, “Bounded and compact vectorial Hankel operators”, Trans. Amer. Math. Soc., 150:2 (1970), 529–539 | DOI | MR | Zbl

[34] V. M. Adamyan, D. Z. Arov, M. G. Krein, “Beskonechnye blochno-gankelevy matritsy i svyazannye s nimi problemy prodolzheniya”, Izv. AN ArmSSR, 6:2–3 (1971), 87–112 | Zbl

[35] H. R. Pousson, “Systems of Toeplitz operators on $H^2$. II”, Trans. Amer. Math. Soc., 133:2 (1968), 527–536 | DOI | MR | Zbl

[36] M. Rabindranathan, “On the inversion of Toeplitz operators”, J. Math. Mech., 19:3 (1969), 195–206 | MR | Zbl

[37] I. B. Simonenko, “Novyi obschii metod issledovaniya lineinykh operatornykh integralnykh uravnenii. I”, Izv. AN SSSR, seriya matem., 29 (1965), 567–586 | MR | Zbl

[38] N. Ya. Krupnik, “Kriterii nëterovosti singulyarnykh integralnykh operatorov s izmerimymi koeffitsientami”, Soobsch. AN GruzSSR, 80:3 (1975), 533–536 | MR | Zbl