Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define $$ \left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|_p = \| |A|^p + |B|^p\|^{1/p}. $$

Q: Is this a norm?

Consider the matrices $C = \left[\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right]$ and $D = \left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right]$. Then $$ \left\|\left[\begin{matrix}C+D \\ D+C\end{matrix}\right]\right\|_p = 2^{1/p}||C+D\| = 2^{1/p+1/2} $$ while $$ \left\|\left[\begin{matrix}C \\ D\end{matrix}\right]\right\|_p + \left\|\left[\begin{matrix}D \\ C\end{matrix}\right]\right\|_p = 2\|C|^p + |D|^p\|^{1/p} = 2\|I\|^{1/p} = 2. $$ Therefore, when $1\leq p< 2$ then $$ \left\|\left[\begin{matrix}C+D \\ D+C\end{matrix}\right]\right\|_p > \left\|\left[\begin{matrix}C \\ D\end{matrix}\right]\right\|_p + \left\|\left[\begin{matrix}D \\ C\end{matrix}\right]\right\|_p $$ and so $\|\cdot\|_p$ is not a norm.

The $p=2$ case does give a norm as $$ \| |A|^2 + |B|^2\|^{1/2} = \left\| [A^* B^*]\left[\begin{matrix}A \\ B\end{matrix}\right] \right\|^{1/2} = \left\|\left[\begin{matrix}A & 0 \\ B & 0\end{matrix}\right] \right\| $$ which allows one to use the triangle inequality for $M_2(B(H))$.

My question is what happens for all of the other cases, $p>2$?