I'm wondering about the problem analogous to the invariant subspace problem, but about fixed points. That is, "Which operators in Banach spaces have fixed points (non-zero points that are mapped to themselves by the operator)?". It seems to me that fixed points and invariant subspaces are at least intuitively similar since they are both fundamental things that map "to themselves".
I know that there are plenty of fixed point theorems that describe the fixed points of certain operators in certain spaces, and I know that various fixed point theorems have been used in the work on invariant subspaces. But I can't find any reference to a complete classification of operators based on their fixed points similar to the goal of the invariant subspace problem.
Is this a question that has been studied much? And if not, why is it less studied than the invariant subspace problem?