# spectral radius of products of symmetric positive definite matrices

I have matrices $$A, B, C, D \in \mathbb R^{n\times n}$$ which are all symmetric positive (semi-)definite.

If I have that

\begin{align} \rho(AB) &< \gamma^2, \\ \rho(CD) &< \gamma^2, \quad \text{and}\\ \rho(BD) &< \gamma^2, \end{align}

can I conclude that $$\rho(AC) < \gamma^2 \tag{?}$$

Here, $$\rho$$ denotes the spectral radius, i.e., the largest in modulo eigenvalue, and $$\gamma \in \mathbb R$$ is a constant.

No. If $$n=1$$ and $$X \ge 0$$, then $$\rho(X)=X.$$
Now let $$A=1, B=3, C=5, D=1/2$$ and $$\gamma =2.$$
$$\rho(AB) < \gamma^2,$$ $$\rho(CD) <\gamma^2$$ and $$\rho(BD) < \gamma^2$$,
but $$\rho(AC) > \gamma^2 .$$