I have Markov semigroups $(P_t)_{t\ge0}$ and $(Q_t)_{t\ge0}$ on a measurable space $(E,\mathcal E)$ with pointwise generators $S$ and $T$, respectively. $S$ and $T$ are related by $$(Sf)(x)=(Tf)(x)+r(x)(Gf)(x)\tag1,$$ where $r:E\to(0,\infty)$ is measurable and $G$ is a bounded linear operator on $F$; the space of bounded measurable real-valued functions on $(E,\mathcal E)$ equipped with the supremum norm.
I would like to investigate the spectral gap of $(P_t)_{t\ge0}$ defined as $$\sup\left\{\Re(\lambda):\lambda\in\sigma(S)\setminus\{0\}\right\}\tag1.$$ My hope is that it can be related (by a formula) to the spectral gap of $(Q_t)_{t\ge0}$.
The answer to this question clearly depends on whether we are considering $(P_t)_{t\ge0}$ and $(Q_t)_{t\ge0}$ as operator semigroups on $F$ or $L^2(\mu)$. Here I'm assuming that $\mu$ is a probability measure on $(E,\mathcal E)$ and $(P_t)_{t\ge0}$ is $\mu$-invariant. $(Q_t)_{t\ge0}$ is not $\mu$-invariant, but I would assume here that it can be extended to $L^2(\mu)$ nevertheless.
I actually don't know how I should approach this. And I'd also like to know if it matters at all whether we consider $F$ or $L^2(\mu)$.
Any help is highly appreciated!