Abstract
We investigate a rearrangement inequality for pairs of n×n matrices: Let \(\|A\|_p\) denote (Tr(A*A)p/2)1/p, the Cp trace norm of an n×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≤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 Lp function spaces, and would show that the unit ball in Cp has the exact same moduli of smoothness and convexity as does the unit ball in Lp 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.
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Almgren, F.J., Lieb, E.H.: Symmetric Decreasing Rearrangement is Sometimes Continuous. J. Am. Math. Soc. 2, 683–773 (1989)
Ball, K., Carlen, E., Lieb, E.H.: Sharp Uniform Convexity and Smoothness Inequalities for Trace Norms. Invent. Math. 115, 463–482 (1994)
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
Epstein, H.: Two Theorems of E. Lieb. Commun. Math. Phys. 31, 317–322 (1973)
Hanner, O.: On the uniform convexity of Lp and ℓp. Ark. Mat. 3, 239–244 (1956)
Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge: Cambridge University Press 1991
Lieb, E.H., Loss, M.: Analysis, 2nd edition. Am. Math. Soc. 2001
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
Tomczak-Jaegermann, N.: The moduli of smootheness and convexity and Rademacher averages of trace classes S p (1≤p<∞). Stud. Math. 50, 163–182 (1974)
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Dedicated to Professor Roberto Conti
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Carlen, E., Lieb, E. Some matrix rearrangement inequalities. Annali di Matematica 185 (Suppl 5), S315–S324 (2006). https://doi.org/10.1007/s10231-004-0147-z
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DOI: https://doi.org/10.1007/s10231-004-0147-z