TY - JOUR
T1 - Inequalities for C-S seminorms and Lieb functions
AU - Horn, Roger A.
AU - Zhan, Xingzhi
PY - 1999/4/15
Y1 - 1999/4/15
N2 - 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.
AB - 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.
UR - https://www.scopus.com/pages/publications/0033439614
U2 - 10.1016/S0024-3795(98)10245-8
DO - 10.1016/S0024-3795(98)10245-8
M3 - 文章
AN - SCOPUS:0033439614
SN - 0024-3795
VL - 291
SP - 103
EP - 113
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 1-3
ER -