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Some matrix rearrangement inequalities. (English) Zbl 1197.15011
Summary: We investigate a rearrangement inequality for pairs of \(n\times n\) matrices: Let \(\|A\|_p\) denote \((\text{Tr}(A^\ast A^{p/2})^{1/p}\), the \(C^p\) trace norm of an \(n\times n\) matrix \(A\). Consider the quantity \(\|A+B\|_p^p+\|A-B\|_p^p\) . Under certain positivity conditions, we show that this is nonincreasing for a natural “rearrangement” of the matrices \(A\) and \(B\) when \(1\leq p\leq 2\). 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 \(L^p\) function spaces, and would show that the unit ball in \(C^p\) has the exact same moduli of smoothness and convexity as does the unit ball in \(L^p\) for all \(1<p<\infty\). At present this is known to be the case only for \(1<p\leq 4/3, p=2\), and \(p\geq 4\). Several other rearrangement inequalities that are of interest in their own right are proved as the lemmas used in proving the main results.

15A45 Miscellaneous inequalities involving matrices
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B20 Geometry and structure of normed linear spaces
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