Timeline for trace norm of AGB, where G is Gaussian random matrix
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 16, 2017 at 12:01 | comment | added | Mikael de la Salle | Non-commutative Khintchine inequalities hold for every $0<p<\infty$, it is just that the formula are different for $p>2$ and $p<2$. But you are right, the inequality $\mathbb E \| \sum_k g_k C_k\|_p \lesssim \max ( \|(\sum C_k C_k^*)^{\frac 1 2}\|_p, \|(\sum C_k^* C_k)^{\frac 1 2}\|_p)$ is true for all $p$ (and trivial if $1 \leq p<2$). | |
| Jan 16, 2017 at 11:31 | comment | added | user58955 | For non-commutative Khintchine, it's just the lower bound for which you need $p\geq 2$, isn't it? The upper bound holds for all $p\geq 1$. (The classical Khintchine does hold for $p<2$ though). | |
| Jan 16, 2017 at 9:08 | history | edited | Mikael de la Salle | CC BY-SA 3.0 |
added 1 character in body
|
| Jan 16, 2017 at 8:54 | comment | added | user58955 | Ah, non-commutative Khintchine -- I didn't think of that -- thanks! | |
| Jan 16, 2017 at 8:28 | history | edited | Mikael de la Salle | CC BY-SA 3.0 |
Answer to the new version of the question
|
| Jan 14, 2017 at 6:18 | comment | added | Mikael de la Salle | No, there is no useful description of the extreme points of the unit ball for the Schatten classes if $p \neq 1,\infty$. However, you can use interpolation to get $(\mathbb E \|A G B\|_p^2)^{\frac 1 2} \leq \min(\|A\|_p \|B\|_2, \|A\|_2 \|B\|_p)$ for every $1\leq p\leq 2$ (the case $p=2$ being an equality, as shown above). | |
| Jan 14, 2017 at 4:08 | comment | added | user58955 | Is there anything similar to "the unit ball of the trace class is the convex hull of the norm 1 rank 1 matrices" for Schatten-p class? That is, if I look at the Schatten p-norm of $AGB$ instead of the trace norm (which is Schatten 1-norm)? | |
| Jan 14, 2017 at 4:04 | vote | accept | user58955 | ||
| Jan 13, 2017 at 19:00 | history | answered | Mikael de la Salle | CC BY-SA 3.0 |