Duality between some linear preservers problems: the invariance of the c-numerical range, the c-numerical radius and certain matrix sets

Let C be a hermitian matrix. We show that the C-numerical range or C-numerical radius preservers are actually the dual operators of some matrix set preservers. We use the known results about the C-numerical range preservers to determine the structure of their dual operators. For the C-numerical radius preservers, we prove a structural theorem for the dual operators, and then determine their structure. This solves a problem in an earlier paper of the authors