  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) $.

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