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+4piint 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.
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