Strongly asymptotically stable Frobenius-Perron operators

Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space and let $T : L^1(X,\Sigma,\mu) \to L^1(X,\Sigma,\mu)$ be a Frobenius-Perron operator.

In 1997 Bartoszek and Brown proved that if $T$ overlaps supports and if there exists $h \in L^1(X,\Sigma,\mu)$, $h > 0$ on $X$, such that $Th = h$, then $T$ is (strongly) asymptotically stable.

In the note we prove that instead of assuming that $h > 0$ on $X$, it is enough to assume that $h\geq 0$ and $h\neq 0$. More precisely, we prove that $T$ is asymptotically stable if and only if $T$ overlaps supports and there exists $h\in L^1(X,\Sigma,\mu)$, $h\geq 0$, $h\neq 0$, such that $Th=h$.