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.
