Daugavet centers and direct sums of Banach spaces

A linear continuous nonzero operator G:X->Y is a Daugavet center if every rank-1 operator T:X->Y satisfies ||G+T||=||G||+||T||. We study the case when either X or Y is a sum $X_1 \oplus_F X_2$ of two Banach spaces $X_1$ and $X_2$ by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces $X_1$ and $X_2$ there exists a Daugavet center acting from $X_1\oplus_F X_2$, and the class of those F such that for some pair of spaces $X_1$ and $X_2$ there is a Daugavet center acting into $X_1\oplus_F X_2$. We also present several examples of such Daugavet centers.