关于$q$-循环算子的范数

摘要：

$q$-对易关系是在Bojeko和Speicher的$q$-Fock空间的基础上建立的，它是在经典玻色子Fock空间和费米子Fock空间上定义的经典对易关系（CCR）和经典反对易关系（CAR）之间插值的.将$q$-Fock空间解释为表现出特定交换性结构的"随机变量"的代数，可以构造所谓的$q$-半圆算子和$q$-循环算子，分别充当经典高斯和复高斯随机变量的$q$-变形。虽然$q$-半圆算子通常被很好地理解，但$q$-循环算子的许多基本性质（特别是其范数的易处理表达式）仍然难以捉摸。受自由概率的组合方法的启发，我们修订了这些运营商的组合配方。指出$q$-半圆的$2n$-范数的有限交和表达式可以通过Touchard在20世纪50年代提出并由Riordan在1974年提炼的弦交叉图的生成函数得到.将这些范数作为$q$中的一个函数推广到复单位球上，并取$n\to\infty$极限，我们恢复了$q$-半圆的范数的熟悉表达式，并证明了在单位球的紧子集上收敛是一致的.相反，$q$-循环的$2n$-范数由奇偶反转的弦交叉图编码，这在组合文献中尚未被表征。我们得出这些对象的某些组合性质，包括封闭形式的表达式的数量，这样的图的任何大小与多达11个交叉点。这些性质使我们能够得出这样的结论：$q$-循环算子的$2n$-范数比$q$-半圆算子的性能差得多。

英文摘要：

The $q$-commutation relations, formulated in the setting of the $q$-Fock space of Bożjeko and Speicher, interpolate between the classical commutation relations (CCR) and the classical anti-commutation relations (CAR) defined on the classical bosonic and fermionic Fock spaces, respectively. Interpreting the $q$-Fock space as an algebra of "random variables" exhibiting a specific commutativity structure, one can construct the so-called $q$-semicircular and $q$-circular operators acting as $q$-deformations of the classical Gaussian and complex Gaussian random variables, respectively. While the $q$-semicircular operator is generally well understood, many basic properties of the $q$-circular operator (in particular, a tractable expression for its norm) remain elusive. Inspired by the combinatorial approach to free probability, we revist the combinatorial formulations of these operators. We point out that a finite alternating-sum expression for $2n$-norm of the $q$-semicircular is available via generating functions of chord-crossing diagrams developed by Touchard in the 1950s and distilled by Riordan in 1974. Extending these norms as a function in $q$ onto the complex unit ball and taking the $n\to\infty$ limit, we recover the familiar expression for the norm of the $q$-semicircular and show that the convergence is uniform on the compact subsets of the unit ball. In contrast, the $2n$-norms of the $q$-circular are encoded by chord-crossing diagrams that are parity-reversing, which have not yet been characterized in the combinatorial literature. We derive certain combinatorial properties of these objects, including closed-form expressions for the number of such diagrams of any size with up to eleven crossings. These properties enable us to conclude that the $2n$-norms of the $q$-circular operator are significantly less well behaved than those of the $q$-semicircular operator.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。