$\mathcal Z$-transform, ROC of a system from dividing to others
I agree with your doubts. The $\mathcal{Z}$-transform

$$\hat{G}(z)=1/G(z)=1-3z^{-1}$$

has the ROC $|z|&gt;0$. The ROC of the multiplication of $H(z)$ and $\hat{G}(z)$ must be the intersection of the ROCs of $H(z)$ and $\hat{G}(z)$ (unless there occur pole-zero cancellations, which is not the case here). Since the ROC of $H(z)$ is $|z|&gt;2$, the ROC of $F(z)=H(z)\hat{G}(z)$ must also be $|z|&gt;2$.

Another way to think about it is that since $H(z)$ and $\hat{G}(z)$ are both causal, also $F(z)$ must be causal, i.e., its ROC must be outside a circle in the complex plane.