Let $M$ be a compact manifold and $\mathcal C^0(M) =\{f:M\to \mathbb R; f\ \text{is continuous}\}$. Suppose that $T:\mathcal C^0(M) \to \mathcal C^0(M)$ a bounded linear operator such that $T$ is a positive operator, i.e. for every $0\leq f\in \mathcal C^0(M)$ we have $0\leq Tf$.
Assume that there exists $h \in \mathcal C^0(M),$ such that $$\lim_{n\to\infty} \|T^n(\mathbf 1) - h\|_{\infty} = 0, \quad \quad (*)$$ where $\mathbf 1 : x \in M \to 1\in \mathbb R.$
Question: I would like to know if hypothesis $(*)$ implies that for every $f\in\mathcal C^0(M),$ the set $$\{T^n f\}_{n\in \mathbb N}, \quad (**)$$ is relative weakly compact.
Does the hypothesis $(*)$ helps to get that $(**)$ is relative weakly compact?