value of the norm of the trace mapping

Question

1) What is the exact value of the norm of the trace mapping ${\rm tr} \colon M_n \to \mathbb{C}$ where we equip $M_n$ with the operator norm $\|A\| = \sup\{\|Ax\| : x\in \ell^2_n \mbox{ with }\|x\|= 1\}$ ?

2) Suppose $1<p<\infty$. Same question if we replace $M_n$ by the finite dimensional Schatten space $S^p_n$.

Remark: if $p=2$, it seems to me that the norm is $\sqrt{n}$.

Answer 1

1) As Zouba said $A\mapsto\text{Tr}(A)$ is positive, so it achieves its norm at the unit. Thus $\|\text{Tr}\|=n$.

2) Assume $\|A\|_p\leq1$. Then in particular $s_j(A)\leq1$ for all singular values of $A$. $$ |\text{Tr}(A)|\leq\text{Tr}(|A|)=\sum_{j=1}^ns_j(A)\leq\left(\sum_{j=1}^ns_j(A)^p\right)^{1/p}\,\left(\sum_{j=1}^ns_j(A)^q\right)^{1/q}\leq n^{1/q}. $$ This upper bound is achieved when $A=n^{-1/p}I$, since $\text{Tr}(n^{-1/p}I)=n^{1-1/p}=n^{1/q}$. Thus $$ \|\text{Tr}\|_p=n^{1-1/p}. $$

In particular, if $p=2$ we get that the norm is $\sqrt n$.

Answer 2

1) The map is positive, hence $\|tr \|=tr(I)=n$.