Integral Equations and Operator Theory

, Volume 77, Issue 2, pp 149–166 | Cite as

Vertical Toeplitz Operators on the Upper Half-Plane and Very Slowly Oscillating Functions

  • Crispin Herrera Yañez
  • Egor A. Maximenko
  • Nikolai Vasilevski


We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend on the imaginary part of the argument only. Such algebra is known to be commutative, and is isometrically isomorphic to an algebra of bounded complex-valued functions on the positive half-line. In the paper we prove that the latter algebra consists of all bounded functions f that are very slowly oscillating in the sense that the composition of f with the exponential function is uniformly continuous or, in other words,
$$\lim_{\frac{x}{y} \to 1} \left|f(x) - f(y)\right| = 0.$$

Mathematics Subject Classification (2010)

Primary 47B35 Secondary 47B32 32A36 44A10 44A15 


Bergman space Toeplitz operators invariant under horizontal shifts Laplace transform very slowly oscillating functions 


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  1. 1.
    Berezin F.A.: Covariant and contravariant symbols of operators. Math. USSR Izvestiya 6, 1117–1151 (1972)CrossRefGoogle Scholar
  2. 2.
    Berezin F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series, and Products. Academic Press, New York (1980)Google Scholar
  4. 4.
    Grudsky S.M., Maximenko E.A., Vasilevski N.L.: Radial Toeplitz operators on the unit ball and slowly oscillating sequences. Comm. Math. Anal. 14, 77–94 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hörmander L.: Estimates for translation invariant operators in L p spaces. Acta Mathematica 104, 93–140 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Rudin W.: Real and Complex Analysis, 3rd Edn. McGraw-Hill, New York (1987)zbMATHGoogle Scholar
  7. 7.
    Schmidt R.: Über divergente Folgen and lineare Mittelbildungen. Math. Z. 22, 89–152 (1925)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Stroethoff K.: The Berezin transform and operators on spaces of analytic functions. Linear Oper. Banach Cent. Publ. 38, 361–380 (1997)MathSciNetGoogle Scholar
  9. 9.
    Suárez D.: Approximation and the n-Berezin transform of operators on the Bergman space. J. Reine Angew. Math. 581, 175–192 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Suárez D.: The eigenvalues of limits of radial Toeplitz operators. Bull. Lond. Math. Soc. 40, 631–641 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Vasilevski N.L.: On Bergman-Toeplitz operators with commutative symbol algebras. Integr. Equ. Oper. Theory 34, 107–126 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Vasilevski N.L.: On the structure of Bergman and poly-Bergman spaces. Integr. Equ. Oper. Theory 33, 471–488 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vasilevski N.L.: Bergman space structure, commutative algebras of Toeplitz operators, and hyperbolic geometry. Integr. Equ. Oper. Theory 46, 235–251 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Vasilevski N.L.: Commutative Algebras of Toeplitz Operators on the Bergman Space, volume 185 of Operator Theory: Advances and Applications. Birkhäuser, Basel (2008)Google Scholar
  15. 15.
    Zorboska N.: The Berezin transform and radial operators. Proc. Am. Math. Soc. 131, 793–800 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Crispin Herrera Yañez
    • 1
  • Egor A. Maximenko
    • 2
  • Nikolai Vasilevski
    • 1
  1. 1.Departamento de MatemáticasCINVESTAVMéxicoMéxico
  2. 2.Escuela Superior de Física y MatemáticasInstituto Politécnico NacionalMéxicoMéxico

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