Commutators and semi-commutators of Toeplitz operators on the Fock space
type: Item
instance of: scholarly article
title: Commutators and semi-commutators of Toeplitz operators on the Fock space
zbMATH Open document ID: 1444.47044
author: Xing-Tang Dong
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2020-04-07T00:00:00Z
review text: Let \[ d\lambda_{\alpha}(z)=\left(\frac{\alpha}{\pi}\right)^ne^{-\alpha|z|^{2}}dv(z) \] be the Gaussian measure on \(\mathbb C^{n}\), where \(n\ge1\), \(\alpha>0\), \(dv(z)\) is the Lebesgue measure. The Fock space \(F^{2}_{\alpha}(\mathbb C^{n})\) is the subspace of entire functions from \(L^{2}(\mathbb C^{n},d\lambda_{\alpha})\). Let \(P_{\alpha}\) be the projector from \(L^{2}(\mathbb C^{n},d\lambda_{\alpha})\) onto \(F^{2}_{\alpha}(\mathbb C^{n})\). For a measurable function \(u\) on \(\mathbb C^{n}\), the Toeplitz operator \(T_{u}\) with the symbol \(u\) is defined by the standard equality \(T_{u}f=P_{\alpha}(uf)\), \(f\in F^{2}_{\alpha}(\mathbb C^{n})\). The author investigates the commutators \(T_{u_{1}}T_{u_{2}}-T_{u_{2}}T_{u_{1}}\) and semi-commutators \(T_{u_{1}}T_{u_{2}}-T_{u_{1}u_{2}}\) of these operators in the case of symbols of the form \(z\mapsto \xi^{p}\bar\xi^{q}|z|^{\mu}\), \(z\in\mathbb C^{n}\), where \(\xi=z/|z|\), \(p,q\in\mathbb N^{n}\), \(\mu\ge0\). Necessary and sufficient conditions are given that the commutator has finite rank. The conditions are formulated in terms of the exponents defining the symbols. A~similar result is obtained for the case of a semi-commutator.
reviewed by: Vladimir S. Pilidi
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 7187889
zbMATH Keywords: Fock space
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: symbol
zbMATH Keywords: finite rank operator
zbMATH Keywords: quasi-homogeneous function
zbMATH Keywords: commutator
zbMATH Keywords: semi-commutator
MaRDI profile type: Publication
full work available at URL: s13226-020-0395-7
OpenAlex ID: W3012224216
cites work: Berezin--Toeplitz quantization and composition formulas
cites work: Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal-Bargmann space
cites work: Algebraic properties and the finite rank problem for Toeplitz operators on the Segal-Bargmann space
cites work: Commuting Toeplitz operators on the Segal-Bargmann space
cites work: On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: Generating subalgebras.
cites work: On the Berezin-Toeplitz calculus
cites work: Mellin transform, monomial symbols, and commuting Toeplitz operators
cites work: Q3097302
cites work: Commutators and semi-commutators of Toeplitz operators on the unit ball
cites work: Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian frames
cites work: The commutants of certain Toeplitz operators on weighted Bergman spaces
cites work: Bicommutants of Toeplitz operators
cites work: Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators
cites work: Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball
cites work: Spaces of Holomorphic Functions in the Unit Ball
DOI: 10.1007/S13226-020-0395-7
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description: scientific article; zbMATH DE number 7187889