Surfaces with $K^2=2\mathcal{X}-2$ and $p_g\geq 5$

This note describes minimal surfaces $S$ of general type satisfying $p_g\geq 5$ and $K^2=2p_g$. For $p_g\geq 8$ the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of such surfaces with non birational canonical map. It turns out that if $p_g\geq 13$, $S$ has always an (unique) genus 2 fibration, whose non 2-connected fibres can be characterized, whilst for $p_g\leq 12$ there are two other classes of such surfaces with non birational canonical map.