A characterization result for compactness of semicommutators of Toeplitz operators

Abstract

In this paper, we investigate the compactness of semicommutators of Toeplitz operators on Hardy spaces and Bergman spaces, focusing on the operators of the form \(T^{H}_{|f|^{2}}-T^{H}_{f}T^{H}_{\overline{f}}\) and \(T^{H}_{|\tilde{f}|^{2}}-T^{H}_{\tilde{f}}T^{H}_{\overline{\tilde{f}}} \), where \(\tilde{f}(z)=f(z^{-1})\). We establish that the compactness of these operators can be characterized through the convergence of the sequence \(\{T^{H}_{n}(|f|^{2})-T^{H}_{n}(f)T^{H}_{n}(\overline{f})\}\) in the sense of singular value clustering. This provides a method for determining the compactness of semicommutators by examining the corresponding Toeplitz matrices derived from the Fourier coefficients of the symbol functions. Furthermore, we identify the function space \(VMO \cap L^{\infty}(\mathbb{T})\) as the largest \(C^{*}\)-subalgebra of \(L^{\infty}(\mathbb{T})\) such that, for any \(f, g \in VMO \cap L^{\infty}(\mathbb{T}) \), sequence \(\{T^{H}_{n}(fg)-T^{H}_{n}(f)T^{H}_{n}(g)\}\) converges in terms of singular value clustering. It is already known that \( VMO \cap L^{\infty}(\mathbb{T})\) is the largest \(C^{*}\)-subalgebra of \(L^{\infty}(\mathbb{T})\) such that, for any \(f, g \in VMO \cap L^{\infty}(\mathbb{T}) \), the operator \(T^{H}_{fg}-T^{H}_{f}T^{H}_{g}\) is compact. Similar considerations are made for Bergman spaces \(A^{2}(\mathbb{D})\), where we obtain partial results. This work links operator theory, numerical linear algebra, and function spaces, providing new insights into the compactness properties of Toeplitz operators and their semicommutators.