Inequalities for C-S seminorms and Lieb functions

Abstract

Let Mn be the space of n × n complex matrices. A seminorm ‖ · ‖ on Mn is said to be a C-S seminorm if ‖A*A‖ = ‖AA*‖ for all A ∈ Mn and ‖A‖≤‖B‖ whenever A, B, and B-A are positive semidefinite. If ‖ · ‖ is any nontrivial C-S seminorm on Mn, we show that ‖∣A‖∣ is a unitarily invariant norm on Mn, which permits many known inequalities for unitarily invariant norms to be generalized to the setting of C-S seminorms. We prove a new inequality for C-S seminorms that includes as special cases inequalities of Bhatia et al., for unitarily invariant norms. Finally, we observe that every C-S seminorm belongs to the larger class of Lieb functions, and we prove some new inequalities for this larger class.