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

TL;DR

This paper extends bounds on the spectral radius of non-negative matrices to real-valued non-negative kernels on measurable spaces, providing inequalities involving an arbitrary kernel and eigenmeasures.

Contribution

It generalizes a recent matrix spectral radius bound to kernels on measurable spaces, broadening applicability in stochastic processes and operator theory.

Findings

Provides bounds on spectral radius for kernels on measurable spaces.

Extends matrix spectral radius results to kernel operators.

Applicable to non-negative kernels with integrability conditions.

Abstract

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) $.