On a characterization of (co)silting objects

We prove that an object $U$ in a triangulated category with coproducts is silting if and only if it is a (weak) generator of the category, the orthogonal class $U^{\perp_{>0}}$ contains $U$, and $U^{\perp_{>0}}$ is closed under direct sums. The proof can be dualized to provide a characterization for cosilting objects in triangulated categories with products.