H-spectral spaces

Let X be a T 0 -space, we say that X is H-spectral if its T 0 -compactification is spectral. This paper deal with topological properties of H-spectral spaces. In the case of T 1 -spaces the T 0 -compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T 1 -space X in order to get its Wallman compactification spectral.