Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $f$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov generators on $\mathcal{C}(X)$ (using the definition of Liggett) with domains $\mathcal{D}_1$ and $\mathcal{D}_2$ such that $\mathcal{D}_1\cap\mathcal{D}_2$ is dense in $\mathcal{C}(X)$.

Question: Is any positive linear combination of $\Omega_1$ and $\Omega_2$ ($\lambda\Omega_1+\mu\Omega_2$ where $\lambda,\mu>0$) a Markov generator?

If this is not true in general, is any of these two hypotheses required:

• $(\Omega_1+\Omega_2)$ is also a Markov generator
• $\Omega_1$ and $\Omega_2$ satisfy both the maximum principle (Proposition 2.2 of Liggett)

Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov generators on $\mathcal{C}(X)$ (using the definition of Liggett) with domains $\mathcal{D}_1$ and $\mathcal{D}_2$ such that $\mathcal{D}_1\cap\mathcal{D}_2$ is dense in $\mathcal{C}(X)$.

Question: Is any positive linear combination of $\Omega_1$ and $\Omega_2$ ($\lambda\Omega_1+\mu\Omega_2$ where $\lambda,\mu>0$) a Markov generator?

If this is not true in general, is any of these two hypotheses required:

• $(\Omega_1+\Omega_2)$ is also a Markov generator
• $\Omega_1$ and $\Omega_2$ satisfy both the maximum principle (Proposition 2.2 of Liggett)