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

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

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 ([email protected])
Jani A. Virtanen
Affiliation:
Department of Mathematics, University of Reading, ReadingRG6 6AX, England ([email protected])
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Abstract

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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
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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

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