I know that eigenvalue estimates involving products of matrices are in general tricky, but probably this question has some hope:
Let $A$ and $B$ be two real symmetric positive semi-definite $n\times n$ matrices (with $A+B$ positive definite if needed). Let $\lambda(X)$ denote any eigenvalue of $X$.
I am fairly sure that $$ |1-\lambda\big( (I+A+B+AB)^{-1}(A+B)\big)| < 1 $$ but I would like ask:
- Do you know a slick proof of this?
- Do you know finer estimates (using,e.g., all eigenvalues/vectors of $A$, $B$ and $A+B$) or techniques that could be helpful?