Bounding the Frobenius norm of a q-deformed commutator

For two n×n complex matrices A and B, we define the q-deformed commutator as [A,B]_q:=AB−qBA 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||_F²≤(1+q²)||A||_F²||B||_F² 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.