  For a given second-order linear elliptic operator $L$ which admits a positive
minimal Green function, and a given positive weight function $W$, we introduce
a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces,
where $1\leq p\leq \infty$. We study some fundamental properties of the
corresponding (weighted) Green operators on these spaces. In particular, we
prove that these Green operators are bounded on $L^p(\phi_p)$ for any $1\leq
p\leq \infty$ with a uniform bound. We study the existence of a principal
eigenfunction for these operators in these spaces, and the simplicity of the
corresponding principal eigenvalue. We also show that such a Green operator is
a resolvent of a densely defined closed operator which is equal to $(-W^{-1})L$
on $C_0^\infty$, and that this closed operator generates a strongly continuous
contraction semigroup. Finally, we prove that if $W$ is a (semi)small
perturbation of $L$, then for any $1\leq p\leq \infty$, the associated Green
operator is compact on $L^p(\phi_p)$, and the corresponding spectrum is
$p$-independent.
