Journal of Operator Theory

Volume 67, Issue 2, Spring 2012 pp. 561-580.

Comparison of matrix norms on bipartite spaces

Authors: Christopher King (1) and Nilufer Koldan (2)
Author institution: (1) Department of Mathematics, Northeastern University, Boston, MA 02115, U.S.A.
(2) Department of Mathematics, Northeastern University, Boston, MA 02115, U.S.A.

Summary: Two non-commutative versions of the classical $L^q(L^p)$ norm on the product matrix algebras ${\mathcal M}_n \otimes {\mathcal M}_m$ are compared. The first norm was defined recently by Carlen and Lieb, as a byproduct of their analysis of certain convex functions on matrix spaces. The second norm was defined by Pisier and others using results from the theory of operator spaces. It is shown that the second norm is upper bounded by a constant multiple of the first for all $1 \leqslant p \leqslant 2$, $q \geqslant 1$. In one case ($2 = p < q$) it is also shown that there is no such lower bound, and hence that the norms are inequivalent. It is conjectured that the norms are inequivalent in all cases.

Keywords: Schatten norm, operator space, monotonicity

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