交换斜Toeplitz算子：多解析与调和符号
Commuting slant Toeplitz operators for polyanalytic and harmonic symbols
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中文总结 AI 辅助
研究单位圆盘Bergman空间上斜Toeplitz算子的交换性与零乘积，证明多解析与调和符号下交换等价于符号线性相关，并给出有限秩交换子为零及乘积为零的充要条件。
中文摘要 AI 辅助
我们研究了单位圆盘Bergman空间上斜Toeplitz算子的交换性与零乘积问题。对于有界多解析符号以及边界值属于$W^{1,2}(\mathbb T)$的有界调和符号，我们证明两个此类算子可交换当且仅当它们的符号在$\mathbb C$上线性相关，且两个符号属于同一类。在多解析情形下，对单个解析分量不施加有界性假设。我们还证明了该类别中每个有限秩交换子均为零，且与非零多解析符号相关的每个有限乘积具有无限秩。最后，对于$s\ge2$，我们证明：若$s$个具有边界值属于$W^{s-1,2}(\mathbb T)$的有界调和符号的斜Toeplitz算子之乘积为零，当且仅当至少有一个符号恒为零。
英文摘要
We study commutativity and zero products of slant Toeplitz operators on the Bergman space of the unit disk. For bounded polyanalytic symbols and for bounded harmonic symbols with boundary values in $W^{1,2}(\mathbb T)$, we prove that two such operators commute if and only if their symbols are linearly dependent over $\mathbb C$, with both symbols belonging to the same class. In the polyanalytic case, no boundedness assumption is imposed on the individual analytic components. We also show that every finite-rank commutator in this class is zero and that every finite product associated with nonzero polyanalytic symbols has infinite rank. Finally, for $s\ge2$, we prove that a product of $s$ slant Toeplitz operators with bounded harmonic symbols whose boundary values belong to $W^{s-1,2}(\mathbb T)$ is zero if and only if at least one symbol vanishes identically.
发表机构
- Liaoning Normal University（辽宁师范大学）
- Dalian University of Technology（大连理工大学）
机构由 AI 辅助整理，请以论文原文为准。
