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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 X1⊕FX2 of two Banach spaces X1 and X2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X1 and X2 there exists a Daugavet center acting from X1⊕FX2, and the class of those F such that for some pair of spaces X1 and X2 there is a Daugavet center acting into X1⊕FX2. We also present several examples of such Daugavet centers.