Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Toeplitz operators with PC symbols
on general Carleson Jordan curves
with arbitrary Muckenhoupt weights


Authors: Albrecht Böttcher and Yuri I. Karlovich
Journal: Trans. Amer. Math. Soc. 351 (1999), 3143-3196
MSC (1991): Primary 47B35; Secondary 30E20, 42A50, 45E05, 47D30
Published electronically: March 29, 1999
MathSciNet review: 1650069
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space $H^p(\Gamma,\omega)$ in case $1<p<\infty $, $\Gamma$ is a Carleson Jordan curve and $\omega$ is a Muckenhoupt weight in $A_p(\Gamma)$. Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve $\Gamma$ and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve $\Gamma$ is nice and $\omega$ is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.


References [Enhancements On Off] (What's this?)

  • 1. A. V. Aĭzenshtat, Yu. I. Karlovich, and G. S. Litvinchuk, Defect numbers of the D. Kveselava–N. Vekua operator with a discontinuous shift derivative, Dokl. Akad. Nauk SSSR 318 (1991), no. 1, 11–16 (Russian); English transl., Soviet Math. Dokl. 43 (1991), no. 3, 633–638 (1992). MR 1122463
  • 2. A. V. Aizenshtat, Yu. I. Karlovich, and G. S. Litvinchuk, The method of conformal gluing for the Haseman boundary value problem on an open contour, Complex Variables 28 (1996), 313-346.
  • 3. Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
  • 4. Albrecht Böttcher and Yuri I. Karlovich, Toeplitz and singular integral operators on Carleson curves with logarithmic whirl points, Integral Equations Operator Theory 22 (1995), no. 2, 127–161. MR 1333286, 10.1007/BF01208347
  • 5. A. Böttcher and Yu. I. Karlovich, Toeplitz and singular integral operators on general Carleson Jordan curves, Singular integral operators and related topics (Tel Aviv, 1995) Oper. Theory Adv. Appl., vol. 90, Birkhäuser, Basel, 1996, pp. 119–152. MR 1413551
  • 6. Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, Springer-Verlag, Berlin, 1990. MR 1071374
  • 7. David W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315–323. MR 0306887
  • 8. L. A. Coburn, Weyl’s theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288. MR 0201969
  • 9. E. A. Danilov, The Riemann boundary value problem on contours with unbounded distortion, Cand. Dissertation, Odessa 1984 [Russian].
  • 10. G. David, L'integrale de Cauchy sur le courbes rectifiables, Prepublication Univ. Paris-Sud, Dept. Math. 82T05, 1982.
  • 11. G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Super 17 (1984), 157-189.
  • 12. Ronald G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 49. MR 0361893
  • 13. E. M. Dynkin, Methods of the theory of singular integrals (Hilbert transform and Calderon-Zygmund theory), In: Itogi Nauki Tekh., Sovr. Probl. Matem., Fund. Napravl., vol. 15, Moscow 1987, pp. 197-292 [Russian]. CMP 20:04
  • 14. E. M. Dyn′kin, Methods of the theory of singular integrals. II. The Littlewood-Paley theory and its applications, Current problems in mathematics. Fundamental directions, Vol. 42 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 105–198, 233 (Russian). MR 1027848
  • 15. E. M. Dynkin and B. P. Osilenker, Weighted norm estimates for singular integrals and their applications, J. Sov. Math. 30 (1985), 2094-2154 [Russian original: Itogi Nauki Tekh., Ser. Mat. Anal. 21 (1983), 42-129].
  • 16. T. Finck, S. Roch, and B. Silbermann, Two projection theorems and symbol calculus for operators with massive local spectra, Math. Nachr. 162 (1993), 167–185. MR 1239584, 10.1002/mana.19931620114
  • 17. John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
  • 18. I. C. Gohberg, On an application of the theory of normed rings to singular integral equations, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 2(48), 149–156 (Russian). MR 0048689
  • 19. I. Gohberg and N. Krupnik, Singular integral operators with piecewise continuous coefficients and their symbols, Math. USSR Izv. 5 (1971), 955-979.
  • 20. I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations, Vols. I and II, Birkhäuser Verlag, Basel, Boston, Berlin 1992 [Russian original: Shtiintsa, Kishinev 1973].
  • 21. I. Gohberg and N. Krupnik, Extension theorems for Fredholm and invertibility symbols, Integral Equations and Operator Theory 16 (1993), 514-529.
  • 22. Israel Gohberg, Naum Krupnik, and Ilya Spitkovsky, Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight, Integral Equations Operator Theory 17 (1993), no. 3, 322–337. MR 1237957, 10.1007/BF01200289
  • 23. E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Coll. Publ., vol. 31, revised edition, Providence, R.I, 1957.
  • 24. Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 0312139, 10.1090/S0002-9947-1973-0312139-8
  • 25. S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
  • 26. V. A. Paatashvili and G. A. Khuskivadze, On the boundedness of the Cauchy singular integral on Lebesgue spaces in the case of non-smooth contours, Trudy Tbilisk. Mat. Inst. AN GSSR 69 (1982), 93-107 [Russian].
  • 27. R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
  • 28. R. K. Seifullayev, The Riemann boundary value problem on non-smooth open curves, Matem. Sb. 112 (1980), 147-161 [Russian] (English transl. in Math. USSR Sb. 40 (1981)).
  • 29. I. B. Simonenko, The Riemann boundary value problem with measurable coefficients, Dokl. Akad. Nauk SSSR 135 (1960), 538-541 [Russian].
  • 30. I. B. Simonenko, Some general questions of the theory of the Riemann boundary value problem, Math. USSR Izv. 2 (1968), 1091-1099.
  • 31. I. B. Simonenko, On the factorization and local factorization of measurable functions, Soviet Math. Dokl. 21 (1980), 271-274.
  • 32. I. B. Simonenko, Stability of weight properties of functions with respect to the singular integral, Matem. Zametki 33 (1983), 409-416 [Russian].
  • 33. Ilya Spitkovsky, Singular integral operators with PC symbols on the spaces with general weights, J. Funct. Anal. 105 (1992), no. 1, 129–143. MR 1156673, 10.1016/0022-1236(92)90075-T
  • 34. Harold Widom, Singular integral equations in 𝐿_{𝑝}, Trans. Amer. Math. Soc. 97 (1960), 131–160. MR 0119064, 10.1090/S0002-9947-1960-0119064-7

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47B35, 30E20, 42A50, 45E05, 47D30

Retrieve articles in all journals with MSC (1991): 47B35, 30E20, 42A50, 45E05, 47D30


Additional Information

Albrecht Böttcher
Affiliation: Faculty of Mathematics, Tech. Univ. Chemnitz-Zwickau, D-09107 Chemnitz, Germany
Email: aboettch@mathematik.tu-chemnitz.de

Yuri I. Karlovich
Affiliation: Faculty of Mathematics, Tech. Univ. Chemnitz-Zwickau, D-09107 Chemnitz, Germany
Address at time of publication: Ukrainian Academy of Sciences, Marine Hydrophysical Institute, Hydroacoustic Department, Preobrazhenskaya Street 3, 270 100 Odessa, Ukraine
Email: karlik@paco.net

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02441-1
Keywords: Carleson condition, Ahlfors--David curve, Muckenhoupt condition, submultiplicative function, Toeplitz operator, singular integral operator, Fredholm operator
Received by editor(s): September 28, 1995
Received by editor(s) in revised form: December 15, 1996
Published electronically: March 29, 1999
Additional Notes: The first author was supported by the Alfried Krupp Förderpreis für junge Hochschullehrer. Both authors were supported by NATO Collaborative Research Grant 950332.
Article copyright: © Copyright 1999 American Mathematical Society