Norm Inequalities for Positive Semidefinite Matrices

In this chapter, we first give a comprehensive answer to a generalized version of the first main Question 3.1.1 for all unitarily invariant norms. We then present several norm inequalities related to the second main Question 3.2.1 , with a particular focus on the trace norm and the Hilbert–Schmidt norm. We provide a complete answer to the second main Question 3.2.1 for the trace norm, as well as a complete answer to a slightly weaker version of this question for the Hilbert–Schmidt norm. Moreover, we discuss majorization relations and unitarily invariant norm inequalities that involve the spectral radius.