A concavity inequality for symmetric norms

We review some convexity inequalities for Hermitian matrices an add one more to the list.Comment: accepted in LA

Total Dilations

(1) Let $A$ be an operator on a space ${\cal H}$ of even finite dimension. Then for some decomposition ${\cal H}={\cal F}\oplus{\cal F}^{\perp}$, the compressions of $A$ onto ${\cal F}$ and ${\cal F}^{\perp}$ are unitarily equivalent. (2) Let $\{A_j\}_{j=0}^n$ be a family of strictly positive operators on a space ${\cal H}$. Then, for some integer $k$, we can dilate each $A_j$ into a positive operator $B_j$ on $\oplus^k{\cal H}$ in such a way that: (i) The operator diagonal of $B_j$ consists of a repetition of $A_j$. (ii) There exist a positive operator $B$ on $\oplus^k{\cal H}$ and an increasing function $f_j : (0,\infty)\longrightarrow(0,\infty)$ such that $B_j=f_j(B)$.Comment: 12 page

An asymmetric Kadison's inequality

Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison's inequality and several operator versions of Chebyshev's inequality. We also discuss well-known results around the matrix geometric mean and connect it with complex interpolation.Comment: To appear in LA

A matrix subadditivity inequality for f(A+B) and f(A)+f(B)

Let f be a non-negative concave function on the positive half-line. Let A and B be two positive matrices. Then, for all symmetric norms, || f(A+B) || is less than || f(A)+f(B) ||. When f is operator concave, this was proved by Ando and Zhan. Our method is simpler. Several related results are presented.Comment: accepted in LA