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:
- weak*-weak*
- strong-weak*
- weak*-strong
- strong-strong
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?
Just from general considerations, condition 3 is the strongest: every open set in the larger (i.e. the strong) topology has an inverse image that is in the smaller topology (i.e. the weak$^\ast$ topology). We just use that we have a map $T$ between a space that has a smaller and a larger topology, no more; assume $T$ is 3-continuous:
It implies 1: if $O$ is open in the smaller topology, it is open in the larger one, so by 3, $T^{-1}[O]$ is open in the smaller one. So it's 1-continuous.
It implies 2: if $O$ is open in the smaller topology, it is open in the larger one, so 3 impies $T^{-1}[O]$ is open in the smaller one, so also in the larger one. So it is 2-continuous.
It implies 4: if $O$ is open in the larger topology, 3 implies that $T^{-1}[O]$ is open in the smaller topology, so also in the larger one, so $T$ is 4-continuous.
Similarly, if $T$ is 4-continuous, it is 2-continuous.
If $T$ is 1-continuous, it is 2-continuous.
I don't see any others that just follow from inclusion of the two topologies. I'm not sure about counterexamples to other implications, my functional analysis is rusty...