Revista Matemática Iberoamericana

Algebras of Toeplitz operators with oscillating symbols

Albrecht Böttcher, Sergei M. Grudsky, and Enrique Ramírez de Arellano

Full-text: Open access


This paper is devoted to Banach algebras generated by Toeplitz operators with strongly oscillating symbols, that is, with symbols of the form $b(e^{i\alpha(x)})$ where $b$ belongs to some algebra of functions on the unit circle and $\alpha$ is a fixed orientation-preserving homeomorphism of the real line onto itself. We prove the existence of certain interesting homomorphisms and establish conditions for the normal solvability, Fredholmness, and invertibility of operators in these algebras.

Article information

Rev. Mat. Iberoamericana, Volume 20, Number 3 (2004), 647-671.

First available in Project Euclid: 27 October 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Secondary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 37C05: Smooth mappings and diffeomorphisms 42A50: Conjugate functions, conjugate series, singular integrals 46H20: Structure, classification of topological algebras 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47L15: Operator algebras with symbol structure

Toeplitz operator Banach algebra $C^*$-algebra Fredholm operator normally solvable operator


Böttcher, Albrecht; Grudsky, Sergei M.; Ramírez de Arellano, Enrique. Algebras of Toeplitz operators with oscillating symbols. Rev. Mat. Iberoamericana 20 (2004), no. 3, 647--671.

Export citation


  • Barría, J. and Halmos, P. R.: Asymptotic Toeplitz operators. Trans. Amer. Math. Soc. 273 (1982), 621-630.
  • Beals, R. and Coifman, R.: Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37 (1984), 39-90.
  • Böttcher, A. and Grudsky, S. M.: Toeplitz operators with discontinuous symbols: phenomena beyond piecewise continuity. In Singular Integral Operators and Related Topics (Tel Aviv, 1995), 55-118. Operator Theory: Advances and Applications 90. Birkhäuser, Basel, 1996.
  • Böttcher, A. and Grudsky, S. M.: On the composition of Muckenhoupt weights and inner functions. J. London Math. Soc. (2) 58 (1998), 172-184.
  • Böttcher, A., Grudsky, S. M. and Spitkovsky, I.: Toeplitz operators with frequency modulated semi-almost periodic symbols. J. Fourier Anal. Appl. 7 (2001), 523-535.
  • Böttcher, A., Grudsky, S. M. and Spitkovsky, I.: Block Toeplitz operators with frequency-modulated semi-almost periodic symbols. Int. J. Math. Math. Sci. 34 (2003), 2157-2176.
  • Böttcher, A. and Silbermann, B.: Analysis of Toeplitz Operators. Springer-Verlag, Berlin, Heidelberg, New York 1990.
  • Böttcher, A. and Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Universitext, Springer-Verlag, New York 1999.
  • Coburn, L. A.: The $C^*$-algebra generated by an isometry. Bull. Amer. Math. Soc. 73 (1967), 722-726.
  • Deift, P.: Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics 3. American Mathematical Society, Providence, RI, 1999.
  • Douglas, R. G.: Banach Algebra Techniques in Operator Theory. Academic Press, New York, 1972 and Springer-Verlag, New York, 1998.
  • Douglas, R. G.: Banach Algebra Techniques in the Theory of Toeplitz Operators. CBMS Regional Conference Series in Mathematics 15. American Mathematical Society, Providence, R. I., 1973.
  • Dybin, V. B. and Grudsky, S. M.: Introduction to the Theory of Toeplitz Operators with Infinite Index. Operator Theory: Advances and Applications 137. Birkhäuser Verlag, Basel, 2002.
  • Fillmore, P. A.: A User's Guide to Operator Algebras. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1996.
  • Garnett, J. B.: Bounded Analytic Functions. Pure and Applied Mathematics 96. Academic Press, New York-London, 1981.
  • Gohberg, I.: On an application of the theory of normed rings to singular integral equations. (Russian). Uspehi Matem. Nauk (N.S.) 7 (1952), 149-156.
  • Gohberg, I. and Feldman, I. A.: Convolution Equations and Projection Methods for their Solution. Translations of Mathematical Monographs 41. American Mathematical Society, Providence, RI, 1974.
  • Gohberg, I. and Krupnik, N.: The algebra generated by the Toeplitz matrices. Funct. Anal. Prilozh. 3 (1969), 119-127.
  • Gohberg, I. and Krupnik, N.: One-Dimensional Linear Singular Integral Equations. I and II. Operator Theory: Advances and Applications 53 and 54. Birkhäuser Verlag, Basel, 1992.
  • Grudsky, S. M.: Singular integral operators with infinite index and Blaschke products. (Russian). Math. Nachr. 129 (1986), 313-331.
  • Grudsky, S. M.: Toeplitz operators and the modelling of oscillating discontinuities with the help of Blaschke products. In Problems and Methods in Mathematical Physics (Chemnitz, 1999), 162-193. Operator Theory: Advances and Applications 121. Birkhäuser, Basel, 2001.
  • Grudsky, S. M., and Khevelev, A. B.: On the invertibility of singular integral operators with periodic coefficients and a shift in $L^2(\bR )$. Soviet Math. Dokl. 18 (1977), 1383-1387.
  • Halmos, P.: A Hilbert Space Problem Book. Van Nostrand, Princeton, 1967.
  • Its, A. R. and Novokshenov, V. Yu.: The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics 1191. Springer-Verlag, Berlin and Heidelberg, 1986.
  • Nikolski, N. K.: Treatise on the Shift Operator. Grundlehren der Mathematischen Wissenschaften 273. Springer-Verlag, Berlin, 1986.
  • Nordgren, E. A.: Composition operators. Canad. J. Math. 20 (1968), 442-449.
  • Roch, S. and Silbermann, B.: Toeplitz-like operators, quasicommutator ideals, numerical analysis I. Math. Nachr. 120 (1985), 141-173.
  • Sarason, D.: Algebras of functions on the unit circle. Bull. Amer. Math. Soc. 79 (1973), 286-299.
  • Sarason, D.: Toeplitz operators with piecewise quasicontinuous symbols. Indiana Univ. Math. J. 26 (1977), 817-838.
  • Shapiro, J. H.: Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.
  • Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants II. Advances in Math. 21 (1976), 1-29.
  • Wolff, T. H.: Two algebras of bounded functions. Duke Math. J. 49 (1982), 321-328.