Complex Analysis and Operator Theory

, Volume 11, Issue 3, pp 707–726 | Cite as

Norm Estimates for Toeplitz Operators on the Bergman Space with Symbols Supported in Circles and Mixed Norms

Article
  • 152 Downloads

Abstract

In this paper we study the class of Toeplitz operators \(T_{\varphi }\), for which \(\int _{0}^{1}\left( 1-t\right) ^{q-1}\left\| T_{\varphi d\sigma _{t}}\right\| _{p}^{q}dt<\infty ,\) for \(1\le p,q<\infty ,\) where \(\varphi \in L^{1}(\mathbb {D}),\) \(\sigma _{t}\) is the Lebesgue measure in the circle \( |\xi |=t\) and \(\left\| \cdot \right\| _{p}\) is the p-Schatten norm of operators defined on the Bergman space of the disc. For that purpose we study the dependence on t of the norm and the p-Schatten norms of Toeplitz operators whose symbols are measures \(\mu \) supported in the circle \(tS^{1}\) with a positive density in \(L^{1}(tS^{1})\).

Keywords

Toeplitz operators Bergman space Berezin Transform Mixed norms Schatten classes 

Mathematics Subject Classification

Primary 47B35 Secondary 46E30 

Notes

Acknowledgments

The authors would like to thank the anonymous reviewer for their valuable comments and suggestions.

References

  1. 1.
    Blasco, O., Pérez-Esteva, S.: Schatten-Herz operators, Berezin transform and mixed norm spaces. Integr. Equat. Oper. Th. 711, 65–90 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Choe, B., Koo, H., Na, K.: Positive Toeplitz operators of Schatten-Herz type. Nagoya Math. J. 185, 31–62 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Diestel, J., Uhl, J.J. Jr.: Vector measures, Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977. xiii+322Google Scholar
  4. 4.
    Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman spaces, Graduate Texts in Mathematics, 199. Springer-Verlag, New York, (2000)Google Scholar
  5. 5.
    Loaiza, M., López-García, M., Pérez-Esteva, S.: Herz classes and Toeplitz operators in the disk. Integr. Equat. Oper. Th. 53(2), 287–296 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality an Oscillatory Integrals. Princeton Univ. Press, (1971)Google Scholar
  7. 7.
    Zhu, K.: Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, 139. Marcel Dekker Inc, New York (1990)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Instituto de Matemáticas, Unidad CuernavacaUniversidad Nacional Autónoma de MéxicoCuernavacaMexico

Personalised recommendations