Let $K$ be a compact set in $\mathbb R^d$. Let $dx$ represent the usual Lebesgue measure and let $Q$ be a compact strictly positive definite integral operator $L^2(K,dx)\to L^2(K,dx)$: $$Q\phi(x) = \int_K Q(x,y)\phi(y)dy\,.$$
Is it possible that there exists a finite borel measure $d\mu$ on $K$ such that $Q$ is not strictly positive definite when defined on $L^2(K,d\mu)$ as $$Q\phi(x) = \int_K Q(x,y)\phi(y)d\mu(y)\,?$$
And what if the integral kernel $(x,y)\mapsto Q(x,y)$ is continuous on $K^2$?