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 $K : \mathrm{E}\times \mathcal{E} \rightarrow \mathbb{R}$, we prove that the spectral radius $\rho(K)$ of $K$ satisfies $$\inf_{x \in \mathrm{E} } \frac{ \mathcal{R} K \cdotp L (x) }{ \mathcal{R} L (x) } \le \rho(K) \le \sup_{x \in \mathrm{E} } \frac{ \mathcal{R} K\cdotp L (x) }{ \mathcal{R} L (x) },$$ where $L$ is an arbitrary Kernel on $(\mathrm{E}, \mathcal{E})$, which is integrable with respect to the left eigenmeasure of $K$ and satisfies $\mathcal{R} L (x) >0$ for all $x \in \mathrm{E}$, and the operator $\mathcal{R}$ is defined by $\mathcal{R}L (x) :=\int_{\mathrm{E}} L(x, \mathrm{d}y)$.