Suppose we have the standard Hilbert-Schmidt integral operator over a closed interval $A$ on a real Hilbert space $H$: $$ Tf(x)=\int_A K(x,y)f(y) dy $$ and $K(x,y)\neq K(y,x)$ so the operator is not self-adjoint. It is well known (since $H$ is real and $T$ is compact), that eigenvalues of $T$ are equivalent to eigenvalues of $T^*$, the adjoint.
Now suppose $\lambda=\lambda_0$ is an eigenvalue of $T$ and the only corresponding eigenfunction $\psi(x)$ is positive. Obviously, it is not true that $\psi(x)$ is necessarily an eigenfunction of $T^*$, but can we say anything about the positivity of the corresponding eigenfunction $\psi^*(x)$? We assume $K$ is not nonnegative.
I guess my main goal of this post is to be directed as to whether the question above is trivial or if it greatly depends on $K$. Thank you.