Let $T_1, \ldots, T_n$ be a set of real-valued symmetric matrices satisfying $Tr(T_j T_k) = 0$ for all $j\neq k$.
Consider the norm $\|T\|_{\infty} = \max_{\|M\|_1 \leq 1} \operatorname{Tr}\left[M^T T\right]$ where $\|M\|_1 = \operatorname{Tr}[\sqrt{M^T M}]$. Note that this is the absolute value of the largest (or smallest) eigenvalue of a symmetric matrix $T$.
If $\|T\|_\infty \leq n$, and $T = \sum_j T_j$ with $T_j$ as above, what does this imply for $\|T_j\|_\infty$? Can we conclude that $\|T_j\|_{\infty} \leq 1$ or some function of $n$?
Note that such a relation would hold, if $\max_{\|M\|_1 \leq 1} \operatorname{Tr}\left[M^T T_j\right] = \max_{\|M\|_1 \leq 1, M \in V} \operatorname{Tr}\left[M^T T_j\right]$ where $V$ is the subspace of symmetric matrices containing $T_j$.