Functiones et Approximatio Commentarii Mathematici

Toeplitz operators with distributional symbols on Fock spaces

Antti Perälä, Jari Taskinen, and Jani Virtanen

Full-text: Open access


We define and study Toeplitz operators $T_a$ with distributional symbols in the setting of weighted Fock spaces of entire functions on the complex plane. Sufficient conditions for boundedness and compactness are presented in terms of the symbol belonging to a weighted Sobolev space $W_\omega^{-m, \infty}$ of negative order.

Article information

Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 203-213.

First available in Project Euclid: 22 June 2011

Permanent link to this document

Digital Object Identifier

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: 46A15

Toeplitz operator Fock space Sobolev space distributional symbol boundedness compactness


Perälä, Antti; Taskinen, Jari; Virtanen, Jani. Toeplitz operators with distributional symbols on Fock spaces. Funct. Approx. Comment. Math. 44 (2011), no. 2, 203--213. doi:10.7169/facm/1308749124.

Export citation


  • R.A. Adams, Sobolev spaces, Academic Press, 1975.
  • A. Alexandrov and G. Rozenblum, Finite rank Toeplitz operators: some extensions of D. Luecking's theorem, J. Funct. Anal. 256 (2009), no. 7, 2291–2303.
  • S. Axler and D. Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (1998), no. 2, 387–400.
  • F.A. Berezin, Quantization, Math. USSR Izv. 8 (1974), 1109–1163.
  • F.A. Berezin, Quantization in complex symmetric spaces, Math. USSR Izv. 9 (1975), 341–379.
  • C.A. Berger and L.A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), no. 2, 819–827.
  • M. Dostanić and K. Zhu, Integral operators induced by the Fock kernel, Integral Eq. Operator Theory 60 (2008), no. 2, 217–236.
  • V. Guillemin, Toeplitz operators in $n$-dimensions, Int. Eq. Oper. Th. 7 (1984), 145–205.
  • R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), 188–254.
  • J. Isralowitch and K. Zhu, Toeplitz operators on the Fock space, Integral Equations Operator Theory 66 (2010), no. 4, 593–611.
  • D.H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), no. 2, 345–368.
  • A. Perälä, J. Taskinen and J. Virtanen, Toeplitz operators with distributional symbols on Bergman spaces, to appear in Proc. Edinburgh Math. Soc.
  • W. Rudin, Functional Analysis, Mc Graw-Hill, 1973.
  • D. Suarez, The essential norm of operators in the Toeplitz algebra on $A\sp p(\ABDAbbB\sb n)$, Indiana Univ. Math. J. 56 (2007), no. 5, 2185–2232.
  • J. Taskinen and J.A. Virtanen, Toeplitz operators on Bergman spaces with locally integrable symbols, Rev. Mat. Iberoam. 26 (2010), no. 2, 693–706.
  • N. Vasilevski, Commutative algebras of Toeplitz operators on the Bergman space, Operator Theory: Advances and Applications, Vol. 185, Birkhäuser Verlag, 2008.
  • K. Zhu, $BMO$ and Hankel operators on Bergman spaces, Pac. J. Math. 155 (1992), no.2, 377–395.
  • K. Zhu, Operator Theory in Function Spaces, 2nd edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007.
  • N. Zorboska, Toeplitz operators with $BMO$ symbols and the Berezin transform, Int. J. Math. Sci. 46 (2003), 2929–2945.