Bounding the Frobenius norm of a q-deformed commutator
Bounding the Frobenius norm of a q-deformed commutator
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DOI：
10.1016/j.laa.2022.03.021
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发表时间：
2022-02
影响因子：
1.1
通讯作者：
D. Chru'sci'nski;G. Kimura;Hiromichi Ohno;Tanmay Singal
中科院分区：
文献类型：
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作者：
D. Chru'sci'nski;G. Kimura;Hiromichi Ohno;Tanmay Singal
对于两个 n× n 复数矩阵 A 和 B，对于实数参数 q，我们将 q 变形换向器定义为 [A, B] q:= A B− q B A。在本文中，我们研究了 Böttcher-Wenzel 不等式的推广，它给出了换向器 (Frobenius) 范数的尖锐上限。在我们的概括中，我们研究了 q 变形换向器的尖锐上限。这种概括可以在两种不同的情况下进行研究：首先是一般矩阵的界限，其次是无痕矩阵。对于这两种情况，都给出了正 q 和负 q 的部分答案和猜想。特别是，用 ||.|| 表示 Frobenius 范数F，当A或B正常时，我们证明以下不等式是正确且尖锐的：||[A, B] q|| F 2 ≤(1+q 2)||一个|| F 2||乙|| F 2 为正 q。此外，我们推测当 A 或 B 无迹时，对于正 q 也有相同的界限。对于负 q，我们推测其他尖锐的上限对于一般场景和 A 或 B 无迹时的场景是正确的。所有猜想都有数字支持，并在 n= 2 时得到证明。
For two n× n complex matrices A and B, we define the q-deformed commutator as [A, B] q:= A B− q B A 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 A or B is normal, we prove the following inequality to be true and sharp:||[A, B] q|| F 2≤(1+ q 2)|| A|| F 2|| B|| F 2 for positive q. Also, we conjecture that the same bound is true for positive q when 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 A or B is traceless. All conjectures are supported with numerics and proved for n= 2.