Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $X\subset B(H,K)$ is called a ternary ring of operators(TRO) provided $xy^*z \in X$ for all $x,y,z \in X$.
On the other hand, an anti-TRO is same as defined above except $-xy^*z \in X$ for all $x,y,z \in X$
Since $X$ is already a subspace it seems both the definitions are same. Can someone please explain the difference?