The unique minimality of an averaging projection

In this paper we will prove that an averaging projection P _a : K (H) → Y, given by the formula P_a(A)=fracA+A^T2P_a(A)=frac{A+A^T}{2} , is the only norm-one projection. Here, K (H) is a space of compact operators on a separable real Hilbert space H, and Y is the subspace of K(H) consisting of all symmetric operators.