Comparison of matrix norms on bipartite spaces

Details Comparison of matrix norms on bipartite spaces Comparison of matrix norms on bipartite spaces

2009-04-10

arxiv.org/abs/0904.1710v2

Mathematics

Functional Analysis

25 pages; result added on non-monotonicity of CL-norm

Scientific paper

Two non-commutative versions of the classical L^q(L^p) norm on the algebra of (mn)x(mn) matrices 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 <= p <= 2, q >= 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.

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