Say $A$ and $B$ are finite-dimensional inner product spaces, and $L$ and $M$ are linear maps from $A$ to $B$.
If there are orthonormal bases $\mathcal{A}$ for $A$ and $\mathcal{B}$ for $B$ such that for every $(a, b) \in \mathcal{A} \times \mathcal{B}$, one of the inner products $\langle b, L a \rangle$ and $\langle b, M a \rangle$ is zero, then $\operatorname{tr}(L^\dagger M) = 0$.
Is the converse true? I suspect it's not, but I'm having a terrible time of proving it.
(I only really care about the answer for complex vector spaces, but counterexamples over other fields would also be appreciated.)
