Inequalities for C-S seminorms and Lieb functions

Let M-n be the space of n x n complex matrices. A seminorm parallel to.parallel to on M-n is said to be a C-S seminorm if parallel to A*A parallel to = parallel to AA*parallel to for all A is an element of M-n and parallel to A parallel to less than or equal to parallel to B parallel to whenever A, B, and B-A are positive semidefinite. If parallel to.parallel to is any nontrivial C-S seminorm on M-n, we show that parallel to\A\parallel to is a unitarily invariant norm on M-n, 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. (C) 1999 Elsevier Science Inc. All rights reserved.