Let $X = [0,1]$ and $K: X \times X \to \mathbb C$ continuous and self-adjoint, meaning that $K(y, x) = \overline {K(x,y)} $. It defines a compact, even Hilbert-Schimdt, self-adjoint convolution operator $T_K$ on $L^2(X)$. Call the eigenvalues $(\lambda_n)_{n \geq 1}$ and orthonormal eigenfunctions $(f_n)_{n \geq 1}$.
Mercer's theorem (Wikipedia) says that when $T_K$ is positive (meaning all $\lambda_n \geq 0$), the series $$\sum_{n \geq 1} \lambda_n f_n(x) \overline {f_n(y)}$$ converges uniformly on $X \times X$ (necessarily to $K(x, y)$, which is the $L^2$-limit).
If have not found any reference that discusses whether the condition that $T_K$ is positive, is necessary.
Can the theorem fail if $T_K$ is not positive?
For a counterexample I don't mind replacing $[0, 1]$ by some other compact Riemannian manifold $X$ with boundary.
