Inequalities for C-S seminorms and Lieb functions Original Research Article

Let Mn be the space of n × n complex matrices. A seminorm double vertical bar · double vertical bar on Mn is said to be a C-S seminorm if double vertical barA*Adouble vertical bar = double vertical barAA*double vertical bar for all A set membership, variant Mn and double vertical barAdouble vertical bar≤double vertical barBdouble vertical bar whenever A, B, and B-A are positive semidefinite. If double vertical bar · double vertical bar is any nontrivial C-S seminorm on Mn, we show that double vertical barmidAdouble vertical barmid 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.