I am working with transfer operators and I reach a point where would be nice if I could use a result from Tosio Kato's book about perturbation theory of linear operators. I think I am able to use Kato's result if the question below has an affirmative answer. I asked this question to some of my colleagues in the math department, but no one was able to help me, so I decided to post it here in hope that some specialist point to me the way to answer it or a reference.

Let $(\Omega,d)$ be a compact metric space, $C(\Omega)$ the space of all real continuous functions and $T:C(\Omega)\to C(\Omega)$ a bounded linear positive operator ($T(f)\geq 0$ if $f\geq 0$) having an isolated positive eigenvalue $\lambda_M$ such that $\lambda_M>\sup\{|\beta|: \beta\in \sigma(T)\setminus\{\lambda_M\} \}$, where $\sigma(T)$ denotes the spectrum of $T$ and suppose additionally that the eigenspace associated to $\lambda_M$ is one-dimensional.

Assume that there is a Borel probability measure $\nu$ over $\Omega$ so that for all $u,v\in C(\Omega)$ we have $\|T(u)-T(v)\|_{L^1(\nu)}\leq \lambda_M\|u-v\|_{L^1(\nu)}$. Since $C(\Omega)$ is dense in $L^1(\nu)$ we can naturally extend $T$ to a bounded linear positive operator on $L^1(\Omega,\mathscr{B}(\Omega),\nu)$, which will be denoted by $\overline{T}$.

Question. Is it true that $\lambda_M$ is an isolated eigenvalue of $\overline{T}$ ?

I am working with transfer operators and I reach a point where would be nice if I could use a result from Tosio Kato's book about perturbation theory of linear operators. I think I am able to use Kato's result if the question below has an affirmative answer. I asked this question to some of my colleagues in the math department, but no one was able to help me, so I decided to post it here in hope that some specialist point to me the way to answer it or a reference.

Let $(\Omega,d)$ be a compact metric space, $C(\Omega)$ the space of all real continuous functions and $T:C(\Omega)\to C(\Omega)$ a bounded linear positive operator ($T(f)\geq 0$ if $f\geq 0$) having an isolated positive eigenvalue $\lambda_M$ such that $\lambda_M>\sup\{|\beta|: \beta\in \sigma(T)\setminus\{\lambda_M\} \}$, where $\sigma(T)$ denotes the spectrum of $T$ and suppose additionally that the eigenspace associated to $\lambda_M$ is one-dimensional.

Assume that there is a Borel probability measure $\nu$ over $\Omega$ so that for all $u\in C(\Omega)$ we have $\|T(u)\|_{L^1(\nu)}\leq \lambda_M\|u\|_{L^1(\nu)}$. Since $C(\Omega)$ is dense in $L^1(\nu)$ we can naturally extend $T$ to a bounded linear positive operator on $L^1(\nu)$, which will be denoted by $\overline{T}$.

Question. Is it true that $\lambda_M>\sup\{|\beta|:\beta\in\sigma(\overline{T})\setminus\{\lambda_M\}\}$ ?