Bounds on the spectral radius of real-valued non-negative Kernels on measurable spaces

In this short technical note, we extend a recently published result [Liao2017] on the Perron root (or the spectral radius) of non-negative matrices to real-valued non-negative kernels on an arbitrary measurable space $(\mathrm{E},\mathcal{E})$. To be precise, for any real-valued non-negative kernel $K:\mathrm{E}\times \mathcal{E}\to \mathbb{R}$, we prove that the spectral radius $\rho (K)$ of $K$ satisfies $$\underset{x\in \mathrm{E}}{\inf }\frac{\mathcal{R}K\cdot L(x)}{\mathcal{R}L(x)}\le \rho (K)\le \underset{x\in \mathrm{E}}{\sup }\frac{\mathcal{R}K\cdot L(x)}{\mathcal{R}L(x)},$$ where $L$ is an arbitrary Kernel on $($\mathrm{E}$, $\mathcal{E}$)$,whichisintegrablewithrespecttothelefteigenmeasureof$K$andsatisfies$ $\mathcal{R}$ L (x) >0 $forall$x \in $\mathrm{E}$$, and the operator $\mathcal{R}$ is defined by $$\mathcal{R}$L (x) :=\int_{$\mathrm{E}$} L(x, $\mathrm{d}$y) $.