Let
- $(E,\mathcal E)$ be a measurable space
- $\kappa$ be a Markov kernel on $(E,\mathcal E)$
- $\mu$ be a probability measure on $(E,\mathcal E)$ reversible with respect to $\kappa$
Under this assumptions, $$\kappa f:=\int\kappa(\;\cdot\;,{\rm d}y)f(y)\;\;\;\text{for }f\in L^2(\mu)$$ is a self-adjoint contraction. Let $$L^2_0:=\left\{f\in L^2(\mu):\int f\:{\rm d}\mu=0\right\},$$ $\kappa_0:=\left.\kappa\right|_{L^2_0}$ and $\sigma(\kappa_0)$ denote the spectrum of $\kappa_0$.
Are we able to show that $\sigma(\kappa_0)\subseteq[-1,1)$.
EDIT: If I remember correctly, $$|\lambda|\le\left\|A\right\|_{\mathfrak L(X)}\;\;\;\text{for all }\lambda\in\sigma(A)$$ for any bounded linear operator $A$ on a $\mathbb R$-Banach space $X$. So, we should immediately obtain $\sigma(\kappa)\subseteq[-1,1]$ and $\sigma(\kappa_0)\subseteq[-1,1]$ by contractivity of $\kappa$.