Suppose that $P_1, \dots, P_k$ are positive $n \times n$ matrices. Is it true that
$$ \sum_{k} \| { P_k } \| \leq n \left\| {\sum_k P_k} \right\|,$$
where $\| \cdot \|$ denotes the operator norm?
My effort:
If the $P_j$ are orthogonal projections then the bound is saturated. I tried to write out the definition $\sum_{k}\langle v, P_k v \rangle $ and then my intuition is that one could somehow group the $P_k$ according to directions. I tried to look this up in the book by Bhatia and some googling.
We have $\|P\|\le\operatorname{tr}(P)\le n\|P\|$ for every $n\times n$ positive semidefinite matrix $P$. Therefore $$ \sum_i\|P_i\|\le\sum_i\operatorname{tr}(P_i)=\operatorname{tr}\left(\sum_iP_i\right)\le n\left\|\sum_iP_i\right\|. $$