Bounding the Frobenius norm of a q-deformed commutator

TL;DR

This paper extends the Böttcher-Wenzel inequality to the $q$-deformed commutator, providing sharp bounds for the Frobenius norm in various matrix scenarios, supported by proofs for $n=2$ and numerical evidence.

Contribution

It introduces a generalized inequality for the $q$-deformed commutator, establishing sharp bounds and conjectures for different matrix types and parameter signs.

Findings

Proved sharp bound for normal matrices with positive $q$.

Conjectured bounds for traceless matrices and negative $q$.

Supported conjectures with numerical experiments and proofs for $n=2$.

Abstract

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 B\"{o}ttcher-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 $n=2$.