发表机构
Department of Mathematics, University of Jordan（约旦大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究径向加权Fock空间上Toeplitz算子生成的$C^*$-代数的交换性，利用表示论刻画了紧子群不变符号情形，并给出特征值积分表示，最后构造反例说明特征值序列的一致闭包性质。
AI 中文摘要
我们研究作用于径向加权Fock空间上的Toeplitz算子。我们利用表示论的工具构造由符号在$\u001bU(n)$作用下不变的Toeplitz算子生成的交换$C^*$-代数族。对于整数$n$的一个分拆$m=(b_1,...,b_k)$，我们将$\u001bU_m:=\u001bU(b_1)\times...\times\u001bU(b_k)$实现为$\u001bU(n)$的块对角子群，并描述加权Fock空间分解为不可约$\u001bU_m$-模的过程。这使我们能够研究符号在$\u001bU_m$下不变的Toeplitz算子以及$k$-拟径向符号，并给出其特征值的显式积分表示。更一般地，对于任意紧子群$H\ub1e4\u001bU(n)$，我们根据表示$\u03c0|_H$的无重性刻画了由$H$-不变Toeplitz算子生成的$C^*$-代数的交换性。最后，对于对数增长的径向权，我们构造一个有界径向符号，其对应的特征值序列关于平方根度量不一致连续。因此，特征值序列集合的一致闭包不等于关于平方根度量一致连续的有界序列的$C^*$-代数。
英文摘要
We study Toeplitz operators acting on radial weighted Fock spaces. We use tools from representation theory to construct commutative families of $C^*$-algebras that are generated by Toeplitz operators whose symbols are invariant under the action of $\U(n)$. For a partition $m=(b_1,...,b_k)$ of an integer $n$, we realize \[\mathbf{\U_m}:=\U(b_1)\times...\times\U(b_k)\] as a block diagonally subgroup of $\U(n)$ and describe the decomposition of the weighted Fock space into irreducible $\mathbf{U_m}$-modules. This allows us to study Toeplitz operators with $\mathbf{U_m}$-invariant, equivalently $k$-quasi-radial, symbols. Also, we provide an explicit integral representation of their eigenvalues. More generally, for an arbitrary compact subgroup $H\subseteq\U(n)$, we characterize the commutativity of the $C^*$-algebra generated by $H$-invariant Toeplitz operators in terms of the multiplicity-free property of the representation $π|_H$. Finally, for a logarithmically growing radial weight, we construct a bounded radial symbol for which the corresponding eigenvalue sequence is not uniformly continuous with respect to the square-root metric. Consequently, the uniform closure of the set of eigenvalue sequences does not coincide with the $C^*$-algebra of bounded sequences that are uniformly continuous with respect to the square-root metric.
