Abstract
AbstractWe 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.
Highlights
The Fock space consists of all holomorphic functions in the n-dimensional complex Euclidean space Cn square-integrable with respect to the Gaussian measure exp(−|z|2) dv, where dv is the Lebesgue measure on Cn
The goal of the present paper is to study the Fredholm properties of Toeplitz operators Tf on generalized Fock spaces in terms of the Berezin transform of f at infinity for symbols of vanishing mean oscillation
We prove that the Toeplitz operator Tf with f ∈ V M O is Fredholm on the generalized Fock space Fφp with 0 < p < ∞ if and only if 0 < lim inf |f (z)| and lim sup |f (z)| < ∞
Summary
The Fock space ( known as the Segal–Bargmann space) consists of all holomorphic functions in the n-dimensional complex Euclidean space Cn square-integrable with respect to the Gaussian measure exp(−|z|2) dv, where dv is the Lebesgue measure on Cn. Its study is genuinely different from other function spaces, such as the Hardy space of the unit circle or the Bergman space of the unit ball, and features unique phenomena that require a distinct set of tools and techniques Some of these applications arise from the theory of operators, such as Toeplitz and Hankel operators, and there is currently considerable interest in the study of these operators on the Fock space and its generalizations, which illustrates c 2019 The Royal Society of Edinburgh. Our result covers Berger and Coburn’s characterization and its recent generalizations to standard weighted Fock spaces Fαp Another important feature of our analysis is the discovery of the second inequality in (1.1), which allows us to treat Toeplitz operators with all symbols. We conclude the paper with a list of open problems related to our work and other important questions about Toeplitz and Hankel operators on Fock spaces
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