 Rank-one operator

For vectors $x,y$ in a Hilbert space, the rank-one operator $(x\otimes y)z=⟨z,y⟩x$ has $\Vert x\otimes y{\Vert }_1=\Vert x\Vert \Vert y\Vert$ and $\mathrm{tr}(x\otimes y)=⟨x,y⟩$.