Commutators and semi-commutators of Toeplitz operators on the Fock space
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.
scientific article; zbMATH DE number 7187889
scientific article