Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-20T02:20:45.128Z Has data issue: false hasContentIssue false
  • English
  • Français

Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents

Part of: Special classes of linear operators Holomorphic functions of several complex variables

Published online by Cambridge University Press:  02 December 2019

Zhangjian Hu  and
Jani A. Virtanen
Zhangjian Hu
Affiliation:
Department of Mathematics, Huzhou University, Huzhou, Zhejiang313000, China (huzj@zjhu.edu.cn)
Jani A. Virtanen
Affiliation:
Department of Mathematics, University of Reading, ReadingRG6 6AX, England (j.a.virtanen@reading.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

Keywords

Toeplitz operatorsFredholm propertiesFock spacesvanishing mean oscillationquasi-Banach spaces

MSC classification

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Secondary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3163 - 3186
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © 2019 The Royal Society of Edinburgh

References

1Al-Qabani, A., Hilberdink, T. and Virtanen, J. A.. Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces. Mathematica Scandinavica (2020).Google Scholar
2Alqabani, A. and Virtanen, J. A.. Fredholm theory of Toeplitz operators on standard weighted Fock spaces. Ann. Acad. Sci. Fenn. Math. 43 (2018), 769–783.CrossRefGoogle Scholar
3Berger, C. A. and Coburn, L. A.. Toeplitz operators on the Segal-Bargmann space. Trans. Amer. Math. Soc. 301 (1987), 813–829.Google Scholar
4Fulsche, R. and Hagger, R.. Fredholmness of Toeplitz Operators on the Fock Space. Complex Anal. Oper. Theory 13 (2019), 375–403.CrossRefGoogle Scholar
5Hagger, R. and Virtanen, J. A.. Compact Hankel operators with bounded symbols, arXiv:1906.09901 (submitted).Google Scholar
6Hu, Z. J. and Lv, X. F.. Toeplitz Operators on Fock Spaces F p(φ). Integral Equ. Operator Theory 80 (2014), 33–59.Google Scholar
7Hu, Z. J., Lv, X. F. and Schuster, A. P.. Bergman spaces with exponential weights. J. Funct. Anal. 276 (2019), 1402–1429.Google Scholar
8Hu, Z. J. and Wang, E. M.. Hankel operators between Fock spaces. Integral Equ. Operator Theory 90 (2018), p. 20.Google Scholar
9Hu, Z. J., Lv, X. F. and Wick, B. D.. Localization and compactness of operators on Fock spaces. J. Math. Anal. Appl. 461 (2018), 1711–1732.CrossRefGoogle Scholar
10Janson, S., Peetre, J. and Rochberg, R.. Hankel forms and the Fock space. Rev. Mat. Iberoamericana 3 (1987), 61–138.CrossRefGoogle Scholar
11Kalton, N. and Mitrea, M.. Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Amer. Math. Soc. 350 (1998), 3903–3922.CrossRefGoogle Scholar
12Lv, X. F.. Hankel operators on Fock spaces F p(φ). Complex Variables Elliptic Equ. 64 (2019), 1522–1533.CrossRefGoogle Scholar
13Runst, T. and Sickel, W.. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations De Gruyter Series in Nonlinear Analysis and Applications, vol. 3 (Berlin: Walter de Gruyter & Co., 1996.Google Scholar
14Range, R. M.. Holomorphic functions and integral representations in several complex variables. Graduate Texts in Mathematics, vol. 108 (New York: Springer-Verlag, 1986).CrossRefGoogle Scholar
15Schuster, A. P. and Varolin, D.. Toeplitz operators and Carleson measures on generalized Bargmann-Fock spaces. Integral Equ. Operator Theory 72 (2012), 363–392.CrossRefGoogle Scholar
16Stroethoff, K..Hankel and Toeplitz operators on the Fock space. Michigan Math. J. 39 (1992), 3–16.Google Scholar
17Wallstén, R.. The S p-criterion for Hankel forms on the Fock space, 0 < p < 1. Math. Scand. 64 (1989), 123–132.CrossRefGoogle Scholar
18Zhu, K. H.. Analysis on Fock spaces. Graduate Texts in Mathematics, vol. 263 (New York: Springer, 2012).CrossRefGoogle Scholar