vector but not matrix norm

Question

Is there a norm $\|\cdot\|_\sharp$ on $\mathbb R^{n\times n}$ for which $\| I\|_\sharp=1$ and there is a matrix $M$ such that $\|M^2\|_\sharp>\|M\|_\sharp^2$?

Answer 1

I'll give an example for $n=3$:

For positive $N$ to be chosen later, define $$ \lVert A\rVert_{\sharp} := \sup_{i,j}\lvert A_{ij}\rvert + N \lvert A_{13}\rvert $$ and let $$ M=\begin{pmatrix} 0&1&0\\ 0&0&1\\ 0&0&0\end{pmatrix}. $$ Then $\lVert M\rVert_{\sharp}=1$ and $\lVert M^2\rVert_{\sharp}=N+1$. Now choose $N$ massive.