Characterizing the Daugavet property by squares of rank-one operators

We prove that, for real Banach spaces of dimension greater than one, each of the identities $\mathrm{\parallel }Id+T^2\mathrm{\parallel }=1+\mathrm{\parallel }{T}^2\mathrm{\parallel }$ and $\mathrm{\parallel }Id-T^2\mathrm{\parallel }=1+\mathrm{\parallel }{T}^2\mathrm{\parallel }$, required for every rank-one operator $T$, characterizes the Daugavet property, thereby answering a question posed by Kadets, Martín, and Merí (2007). Consequently, every extremely non-complex Banach space of dimension greater than one has the Daugavet property, answering a question of Martín and Merí (2011). We also compare pointwise versions of the square Daugavet properties with Daugavet points, $\mathrm{\Delta }$-points, and $\mathrm{\nabla }$-points.