c-Numerical radius isometries on matrix algebras and triangular matrix algebras

Let c=(c1,…,cn)t∈Rn and Mn be the set of n×n complex matrices. For any A∈Mn, define the c-numerical range and the c-numerical radius of A byWc(A)={∑i=1nci〈Axi,xi〉:{x1,…,xn}is an orthonormal set in Cn} andwc(A)=max⁡{|z|:z∈Wc(A)}, respectively. Let Tn be the set of n×n upper triangular matrices. When wc(⋅) is a norm on Mn, mappings T on Mn (or Tn) satisfyingwc(T(A)−T(B))=wc(A−B) for all A, B are characterized. As an intermediate step, we also characterize additive c-numerical range preservers on Mn (or Tn).