Skip to main content
SpringerLink
Log in

Toeplitz and Related Operators on Polyanalytic Fock Spaces

  • Chapter
  • First Online:
Toeplitz Operators and Random Matrices

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 289))

Abstract

We give a characterization of compact and Fredholm operators on polyanalytic Fock spaces in terms of limit operators. As an application we obtain a generalization of the Bauer–Isralowitz theorem using a matrix valued Berezin type transform. We then apply this theorem to Toeplitz and Hankel operators to obtain necessary and sufficient conditions for compactness. As it turns out, whether or not a Toeplitz or Hankel operator is compact does not depend on the polyanalytic order. For Hankel operators this even holds on the true polyanalytic Fock spaces.

In Memory of Harold Widom

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. L.D. Abreu, H. Feichtinger: Function spaces of polyanalytic functions, in Harmonic and Complex Analysis and its Applications, Trends Math. (Birkhäuser, Basel, 2014), pp. 1–38

    Google Scholar 

  2. S. Axler, D. Zheng, Compact operators via the Berezin transform. Indiana Univ. Math. J. 47(2), 387–400 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.B. Balk, Polyanalytic functions and their generalizations, in Complex Analysis, I, Encyclopaedia Math. Sci., vol. 85 (Springer, Berlin, 1997), pp. 195–253

    Google Scholar 

  4. W. Bauer, R. Fulsche, Berger-Coburn theorem, localized operators, and the Toeplitz algebra, in Operator Algebras, Toeplitz Operators and Related Topics, ed. by W. Bauer, R. Duduchava, S. Grudsky, M. Kaashoek, Oper. Theory Adv. Appl., vol. 279 (Birkhäuser, 2020)

    Google Scholar 

  5. W. Bauer, J. Isralowitz, Compactness characterization of operators in the Toeplitz algebra of the Fock space \(F^p_{\alpha }\). J. Funct. Anal. 263, 1323–1355 (2012)

    Google Scholar 

  6. W. Bauer, L.A. Coburn, R. Hagger, Toeplitz quantization on Fock space. J. Funct. Anal. 274, 3531–3551 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. C.A. Berger, L.A. Coburn, Toeplitz operators on the Segal-Bargmann space. Trans. Am. Math. Soc. 301(2), 813–829 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Engliš, Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integral Equ. Oper. Theory 33(4), 426–455 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Fulsche, R. Hagger, Fredholmness of Toeplitz operators on the Fock space. Complex Anal. Oper. Theory 13, 375–403 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Hagger, The essential spectrum of Toeplitz operators on the unit ball. Integral Equ. Oper. Theory 89(4), 519–556 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Hagger, Limit operators, compactness and essential spectra on bounded symmetric domains. J. Math. Anal. Appl. 470(1), 470–499 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Hagger, C. Seifert, Limit operators techniques on general metric measure spaces of bounded geometry. J. Math. Anal. Appl. 489(2), 124180, 36 pp. (2020)

    Google Scholar 

  13. R. Hagger, J.A. Virtanen, Compact Hankel operators with bounded symbols. J. Operat. Theor. 86(2), 317–329 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Isralowitz, M. Mitkovski, B. Wick, Localization and compactness in Bergman and Fock spaces. Indiana Univ. Math. J. 64, 1553–1573 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. G.V. Kolossov, Sure les problèmes d’élasticité à deux dimensions. C. R. Acad. Sci. 146, 522–525 (1908)

    Google Scholar 

  16. P. Krajkiewicz, The Picard theorem for multianalytic functions. Pacific J. Math. 48, 423–439 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Lindner, Infinite Matrices and Their Finite Sections (Birkhäuser Verlag, Basel, 2006)

    MATH  Google Scholar 

  18. F. Luef, E. Skrettingland, A Wiener Tauberian theorem for operators and functions. J. Funct. Anal. 280, 108883, 44 pp. (2021)

    Google Scholar 

  19. M. Mitkovski, B. Wick: A Reproducing Kernel Thesis for Operators on Bergman-type Function Spaces, Oper. Theory Adv. Appl., vol. 117 (Birkhäuser, Basel, 2000), pp. 371–386

    Google Scholar 

  20. N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Fundamental Equations, Plane Theory of Elasticity, Torsion and Bending, Translated by J. R. M. Radok (P. Noordhoff, Ltd., Groningen, 1963)

    Google Scholar 

  21. V.S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory, Operator Theory: Advances and Applications, vol. 150 (Birkhäuser, Basel, 2004)

    Google Scholar 

  22. G. Rozenblum, N. Vasilevski, Toeplitz operators in polyanalytic Bergman type spaces, in Functional Analysis and Geometry: Selim Grigorievich Krein Centennial, Contemp. Math., vol. 733 (Amer. Math. Soc., Providence, 2019), pp. 273–290

    Google Scholar 

  23. N. Vasilevski, Poly-Fock spaces, in Differential Operators and Related Topics, Vol. I (Odessa, 1997), Oper. Theory Adv. Appl., vol. 117 (Birkhäuser, Basel, 2000), pp. 371–386

    Google Scholar 

  24. H. Wolf, Toeplitz operators on polyanalytic functions and Klee’s combinatorial identity. Math. Nachr. 166, 5–15 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Xia, Localization and the Toeplitz algebra on the Bergman space. J. Funct. Anal. 269, 781–814 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Xia, D. Zheng, Localization and Berezin transform on the Fock sapce. J. Funct. Anal. 264, 97–117 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. K. Zhu, Analysis on Fock Spaces, Grad. Texts in Math., vol. 263 (Springer, New York, 2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

    Raffael Hagger

Authors
  1. Raffael Hagger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raffael Hagger .

Editor information

Editors and Affiliations

  1. American Institute of Mathematics, Palo Alto, CA, USA

    Estelle Basor

  2. Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany

    Albrecht Böttcher

  3. Department of Mathematics, University of California, Santa Cruz, CA, USA

    Torsten Ehrhardt

  4. Department of Mathematics, University of California Davis, Davis, CA, USA

    Craig A. Tracy

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Hagger, R. (2022). Toeplitz and Related Operators on Polyanalytic Fock Spaces. In: Basor, E., Böttcher, A., Ehrhardt, T., Tracy, C.A. (eds) Toeplitz Operators and Random Matrices. Operator Theory: Advances and Applications, vol 289. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-13851-5_18

Download citation

Publish with us

Policies and ethics

Search

Navigation