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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.

##### MSC:
 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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##### References:
 [1] Almgren, J. Am. Math. Soc., 2, 683 (1989) [2] Ball, Invent. Math., 115, 463 (1994) · Zbl 0803.47037 [3] Carlen, E., Lieb, E.H.: A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy. Advances in the Mathematical Sciences, AMS Translations, 189, Series 2 (1999), pp. 59-68. Also in Inequalities, Selecta of Elliott H. Lieb, ed. by M. Loss, M.B. Ruskai. Springer 2002 · Zbl 0933.47014 [4] Epstein, Lieb. Commun. Math. Phys., 31, 317 (1973) · Zbl 0257.46089 [5] Hanner, Ark. Mat., 3, 239 (1956) [6] Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge: Cambridge University Press 1991 · Zbl 0729.15001 [7] Lieb, E.H., Loss, M.: Analysis, 2nd edition. Am. Math. Soc. 2001 [8] Lieb, E.H., Thirring, W.: Inequalities for the Moments of the eigenvalues of the Schrodinger Hamiltonian and Their Realtion to Sobolev Inequalities. In: Studies in Mathematical Physics, pp. 269-303, ed. by E. Lieb. B. Simon, A. Wightman. Princeton: Princeton University Press 1976 [9] Tomczak-Jaegermann, Stud. Math., 50, 163 (1974)
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