Let us have a mapping $T:X^*\to X^*$. We can endow domain and codomain with norm 'strong' topology (let X be Banach space), or weak star topology. That gives us total 4 combinations:
From wikipedia I have concluded, that the continuity of T in 3 implies the continuity of T in 1 and also $3\implies 4$, and it seems reasonable to me.
If there are more such relations, or some small conditions that will generate such? I was not able to find much concrete stuff on weak star continuity, namely examples or conditions in usual spaces as $l_p$ and others. Do you know some materials with such characterizations?