H-spectral spaces

Let X be a T0-space, we say that X is H-spectral if its T0-compactification is spectral. This paper deal with topological properties of H-spectral spaces. In the case of T1-spaces the T0-compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T1-space X in order to get its Wallman compactification spectral.