I'm searching for a textbook studying the "spectral gap" $\operatorname{gap}A$ of a bounded linear operator $A$ on a $\mathbb R$-Hilbert space $H$.
I've encountered this notion in the following setting: Let $(E,\mathcal E,\mu)$ be a probability space, $\kappa$ be a Markov kernel on $(E,\mathcal E)$ which is symmetric with respect to $\mu$ and hence $$\kappa f:=\int\kappa(\;\cdot\;,{\rm d}y)f(y)\;\;\;\text{for }f\in L^2(\mu)$$ is a nonnegative self-adjoint operator on $L^2(\mu)$. Note that $1$ is an eigenvalue of $\kappa$ and $${\mathcal N(\lambda-\kappa)}^\perp=L^2_0(\mu):=\left\{f\in L^2(\mu):\int f\:{\rm d}\mu=0\right\}.$$ Let $\kappa_0:=\left.\kappa\right|_{L^2_0(\mu)}$. Note that $\left\|\kappa_0\right\|_{L^2_0(\mu)}$ is equal to the spectral radius $r(\kappa_0)$ of $\kappa_0$. In this context, the spectral gap of $\mathcal L:=\kappa_0-\operatorname{id}_{L^2_0(\mu)}$ is defined by $$\operatorname{gap}\mathcal L:=\inf_{f\perp_{L^2(\mu)}1}\frac{\langle -Lf,f\rangle_{L^2(\mu)}}{\left\|f\right\|_{L^2(\mu)}^2}$$ and is claimed to equal $\inf\sigma(-L)$. It is said that $\operatorname{gap}\mathcal L$ is the gap in the spectrum of $-L$ between the eigenvalue $0$ and the infimum of the spectrum on the complement of the corresponding eigenspace.
I'd be interested in such considerations in a more general context. Does anybody know a reference?