  To determine the stability and instability of a given steady galaxy
configuration is one of the fundamental problems in the Vlasov theory for
galaxy dynamics. In this article, we study the stability of isotropic spherical
symmetric galaxy models $f_{0}(E)$, for which the distribution function $f_{0}$
depends on the particle energy $E$ only. In the first part of the article, we
derive the first sufficient criterion for linear instability of $f_{0}(E):$
$f_{0}(E)$ is linearly unstable if the second-order operator \[
A_{0}\equiv-\Delta+4\pi\int f_{0}^{\prime}(E)\{I-\mathcal{P}\}dv \] has a
negative direction, where $\mathcal{P}$ is the projection onto the function
space $\{g(E,L)\},$ $L$ being the angular momentum [see the explicit formula
(\ref{A0-radial})]. In the second part of the article, we prove that for the
important King model, the corresponding $A_{0}$ is positive definite. Such a
positivity leads to the nonlinear stability of the King model under all
spherically symmetric perturbations.
