Some Matrix Rearrangement Inequalities

We investigate a rearrangement inequality for pairs of n × n matrices: Let ‖A‖p denote (Tr(A∗A)p/2)1/p, the C trace norm of an n×n matrix A. Consider the quantity ‖A+B‖p+‖A−B‖p. Under certain positivity conditions, we show that this is nonincreasing for a natural “rearrangement” of the matrices A and B when 1 ≤ p ≤ 2. We conjecture that this is true in general, without any restrictions on A and B. Were this the case, it would prove the analog of Hanner’s inequality for L function spaces, and would show that the unit ball in C has the exact same moduli of smoothness and convexity as does the unit ball in L for all 1 < p < ∞. At present this is known to be the case only for 1 < p ≤ 4/3, p = 2, and p ≥ 4. Several other rearrangement inequalities that are of interest in their own right are proved as the lemmas used in proving the main results.