# Potential p-norm on tuples of positive operators

This is a follow-up to this question on p-norms of tuples of operators.

Consider $$\left[\begin{matrix} A \\ B \end{matrix}\right] \in B(H)^2_+$$, meaning $$A,B\geq 0$$, and define $$\left\|\left[\begin{matrix} A \\ B \end{matrix}\right]\right\|_p = \|(A)^p + (B)^p\|^{1/p}.$$ The previous question's answer conclusively showed that the generalization of the above is a norm only when $$p=2$$. However, the proof does not immediately transfer to this more limited situation (at least I can't see how to do it).

In this new case, it is obvious that this satisfies the triangle inequality on the positive cone for $$p=1$$ and $$p=2$$.

Q: For which $$p$$ does $$\|\cdot\|_p$$ satisfy the triangle inequality on $$B(H)^2_+$$?

If the answer is "no" for some $$p$$ then I would be very interested in a specific example as well as a more general result.

Edit: If this is established for any given p then one can create a norm via Haagerup's decomposition ideas.