if $||AB||_\infty<1$ is I-AB is positive definite ? provided A,B are symmetric and A+B is positive definite

Question

I have $||AB||_\infty <1 $ (and hence $\rho(AB)<1$), where A,B are the diagonal and off-diagonal components of some symmetric positive definite matrix $M \ni M=A+B$. Can it be concluded that $I-AB$ is positive definite ?

my work: $\lambda(I-AB)=1-\lambda(AB)$, since $\rho(AB)<1$ if it can be show that $AB$ is positive definite then $\lambda(I-AB)>0$. But how to show $AB$ is positive definite ?