论文标题
Toeplitz操作员半公路的紧凑性 - 表征
Compactness of Semicommutators of Toeplitz operators -- a Characterization
论文作者
论文摘要
令$ t_ {f} $表示hardy空间上的toeplitz运算符$ h^{2}（\ mathbb {t}）$，让$ t_ {n}（f）$为相应的$ n \ times n $ n $ toeplitz matrix。在本文中，我们表征运算符的紧凑性$ t_ {| f |^{2}} -T_ {f}美元$ \ {t_ {n}（| f |^{2}） - t_ {n}（f）T_ {n}（\ overline {f}）\} $在奇异值集群的意义上。因此，我们获得了一种使用从符号函数的傅立叶系数（Toeplitz矩阵）获得的矩阵来检查Toeplitz运算符的半公路的紧凑性。函数空间$ vmo \ cap l^{\ infty}（\ mathbb {t}）$是最大的$ c^{*} $ - $ l^{\ infty}（\ mathbb {t}）的$ l^{\ mathbb {T l^{\ infty}（\ mathbb {t}）$，$ t_ {fg} -t_ {f} t_ {g} $是紧凑的。在本文中，我们从$ \ {t_ {n}（fg） - t_ {n}（n}（f）t_ {n}（n}（n}（g）\} $中，在ossing ossing ossing osing ossing sing olar valueal clessing clistersing中，我们都会获得$ vmo \ cap l^{\ mathbb {t}）$的特征。确切地说，$ vmo \ cap l^{\ infty}（\ mathbb {t}）$是最大的$ c^{*} $ - $ l^{\ infty}（\ mathbb {t}）的subalgebra of $ l^{\ mathbb {t}）$，每当$ f，g \ in vmo \ cap cap cap cap p， l^{\ infty}（\ mathbb {t}）$，$ \ {t_ {n}（fg）（fg）-t_ {n}（f）t_ {n}（n}（g）\} $从单数值群集的感觉中收敛。
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 $C^{*}$-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 $C^{*}$-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.
