论文标题
界限Q授权的换向器的frobenius规范
Bounding the Frobenius norm of a q-deformed commutator
论文作者
论文摘要
对于两个$ n \ times n $复杂矩阵$ a $和$ b $，我们将$ q $ formed的换向器定义为$ [a，b] _q：= a b -q ba $用于真实参数$ q $。在本文中，我们研究了Böttcher-Wenzel不平等的概括，这给出了换向因子（Frobenius）规范的急剧上限。在我们的概括中，我们研究了$ q $ podformed的换向器上的急剧上限。可以在两种不同的情况下研究这种概括：首先是一般矩阵的界限，其次是无可读矩阵的范围。对于这两种情况，都以正面和负$ Q $的方式给出了部分答案和猜想。特别是，用$ ||。|| _f $表示Frobenius Narm，当$ a $或$ b $是正常的时，我们证明以下不平等是真实而敏锐的：$ || [a，b] _q || _f^2 \ le \ left（1+q^2 \ right）|| a || _f^2 || b || b || _f^2 $ for -osutial $ q $。另外，我们猜测，当$ a $ a $ a $ a $ a $ $ a $无可用时，正面$ q $是正确的。对于负$ Q $，我们猜测对于通用场景和$ a $ a或$ b $的一般场景和场景是正确的。所有猜想均以数字支持，并以$ n = 2 $证明。
For two $n \times n$ complex matrices $A$ and $B$, we define the $q$-deformed commutator as $[ A, B ]_q := A B - q BA$ for a real parameter $q$. In this paper, we investigate a generalization of the Böttcher-Wenzel inequality which gives the sharp upper bound of the (Frobenius) norm of the commutator. In our generalisation, we investigate sharp upper bounds on the $q$-deformed commutator. This generalization can be studied in two different scenarios: firstly bounds for general matrices, and secondly for traceless matrices. For both scenarios, partial answers and conjectures are given for positive and negative $q$. In particular, denoting the Frobenius norm by $||.||_F$, when either $A$ or $B$ is normal, we prove the following inequality to be true and sharp: $|| [ A , B ]_q||_F^2 \le \left(1+q^2 \right) ||A||_F^2 ||B||_F^2$ for positive $q$. Also, we conjecture that the same bound is true for positive $q$ when either $A$ or $B$ is traceless. For negative $q$, we conjecture other sharp upper bounds to be true for the generic scenarios and the scenario when either of $A$ or $B$ is traceless. All conjectures are supported with numerics and proved for $n=2$.
