Jean-Christophe Bourin

Some inequalities for norms on matrices and operators

We study several inequalities for norms on matrices, in particular for the Hilbert–Schmidt and operator norms. These inequalities occur when comparing norms of the products XY and YX for matrices X and Y with suitable assumptions. we also point out some trace inequalities.

Unitary orbits of Hermitian operators with convex or concave functions

This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen, sub or super-additivity type inequalities are considered. Some of them are substitutes to classical inequalities (Choi, Davis, Hansen-Pedersen) for operator convex or concave functions. Various trace, norm and determinantal inequalities are derived. Combined with an interesting decomposition for positive semi-definite matrices, several results for partitioned matrices are also obtained.

MATRIX SUBADDITIVITY INEQUALITIES AND BLOCK-MATRICES

We give a number of subadditivity results and conjectures for symmetric norms, matrices and block-matrices. Let A, B, Z be matrices of same size and suppose that A, B are normal and Z is expansive, i.e. Z*Z ≥ I. We conjecture that for all non-negative concave function f on [0,∞) and all symmetric norms ‖ · ‖ (in particular for all Schatten p-norms). This would extend known results for positive operator to all normal operators. We prove these inequalities in several cases and we propose some related open questions, both in the positive and normal cases. As nice applications of subadditivity results we get some unusual estimates for partitioned matrices. For instance, for all symmetric norms and 0 ≤ p ≤ 1, whenever the partitioned matrix is Hermitian or its entries are normal. We conjecture that this estimate for f(t) = tp remains true for all non-negative concave functions f on the positive half-line. Some results for general block-matrices are also given.

NORM AND ANTI-NORM INEQUALITIES FOR POSITIVE SEMI-DEFINITE MATRICES

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if

A MATRIX SUBADDITIVITY INEQUALITY FOR SYMMETRIC NORMS

g(t)=\sum_{k=0}^m a_kt^k

DECOMPOSITION AND PARTIAL TRACE OF POSITIVE MATRICES WITH HERMITIAN BLOCKS

is a polynomial of degree

Singular values of compressions, restrictions and dilations

m

Matrix inequalities from a two variables functional

with non-negative coefficients, then, for all positive operators

Anti-norms on Finite von Neumann Algebras

A,\,B

Subadditivity Inequalities for Compact Operators

and all symmetric norms,