I am working with an operator $L$ defined as follows, $$(Lf)(x',y') = \int_{x,y}f(x,y)g(x,y)T(x',y'\vert x,y) dxdy,$$ where $x,y,x',y' \in \mathbb R$, functions $f,g: \mathbb R^2 \rightarrow \mathbb R^2$ and $T:\mathbb R^4 \rightarrow \mathbb R$.
Additionally, $g(x,y)$ is some given function such that $g(x,y) \in [0,1)$ for all $x,y$, and $T(x',y'\vert x,y)$ is a (well-defined continuous) transition probability density (i.e. it gives the probability density of moving from states $x,y$ in this period to $x',y'$ in the next period).
I want to know if under the conditions provided for $g,T$ above, whether the operator $I-L$ is invertible (where $I$ is the identity operator).