Compactness of Semicommutators of Toeplitz operators -- a Characterization

Abstract

Let $T_f$ denote the Toeplitz operator on the Hardy space $H^2(\mathbb{T})$ and let $T_n(f)$ be the corresponding $n\times n$ Toeplitz matrix. In this paper, we characterize the compactness of the operators $T_{|f{|}^2}-T_f{T}_{\overline{f}}$ and $T_{|\tilde{f}{|}^2}-T_{\tilde{f}}{T}_{\overline{\tilde{f}}},$ where $\tilde{f}(z)=f(z^{-1}),$ in terms of the convergence of the sequence $\{T_n(|f{|}^2)-T_n(f)T_n(\overline{f})\}$ in the sense of singular value clustering. Hence we obtain a method to check the compactness of semicommutators of Toeplitz operators using the matrices obtained from the Fourier coefficients of the symbol function (Toeplitz matrices). The function space $VMO\cap L^{\infty }(\mathbb{T})$ is the largest $C^{\ast }$-subalgebra of $L^{\infty }(\mathbb{T})$ with the property that whenever $f,g\in VMO\cap L^{\infty }(\mathbb{T})$, $T_{fg}-T_f{T}_g$ is compact. In this article, we obtain a characterization of $VMO\cap L^{\infty }(\mathbb{T})$ in terms of the convergence of $\{T_n(fg)-T_n(f)T_n(g)\}$ in the sense of singular value clustering. To be precise, $VMO\cap L^{\infty }(\mathbb{T})$ is the largest $C^{\ast }$-subalgebra of $L^{\infty }(\mathbb{T})$ with the property that whenever $f,g\in VMO\cap L^{\infty }(\mathbb{T})$, $\{T_n(fg)-T_n(f)T_n(g)\}$ converges in the sense of singular value clustering.