Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ||A||_{op}||B||_{op}$ since the operator norm is sub-multiplicative. Let us fix the operator norm to be the 2-norm or the one induced by the Euclidean vector norm.
I would like to increase this gap using a signature matrix $\tilde{I}$. Intuitively, I am just negating certain columns of $A$ (or equivalently, certain rows of $B$).
How small can we make the ratio (by a good choice of $\tilde{I}$):
\begin{equation}\frac{||A\tilde{I}B||_{op}}{||A||_{op}||B||_{op}}\end{equation}
Moreover, is there a constant $C$ such that \begin{equation}\frac{||A\tilde{I}B||_{op}}{||A||_{op}||B||_{op}} \leq C\end{equation} and \begin{equation}\frac{||B\tilde{I}A||_{op}}{||B||_{op}||A||_{op}} \leq C\end{equation} and can we find such an $\tilde{I}$?
Ideally I hope for a bound around $\tan^2(\pi/8)$ :) Is this part of a general theory of trying to "almost orthogonalize" matrices and are there any good relevant references? I came across Grothendeick's Inequality while searching for relevant info, but not sure how related it is.
Edit: If $A$ and $B$ are both identity matrices for instance (or diagonal matrices), clearly nothing can be done. The context for this question is that the adjacency matrix of a d-regular graph has been split into the adjacency matrices of two d/2-regular edge-disjoint spanning subgraphs. The question might be more meaningful in this context.
Clearly, it cannot be a constant for all $A$ and $B$ but in this context, it would help if the bound depends on $||A+B||,||A||$ and $||B||$.
If $A$ and $B$ are diagonal matrices, the signature matrix doesn't affect $\|A \tilde{I} B \|$.