Let $H$ and $K$ be Hilbert spaces and $X=B(H,K)$ denotes the spaces of bounded linear operators. For $T_1, T_2$ in $X$, we define $D_{T_1,T_2}:X \to X$ as $D_{T_1,T_2}(T)=TT_1^*T_2$. Finally define $V^0=$span$\{D_{T_1,T_2}, T_1, T_2 \in X\}$. One can check that $V^0$ is pre $C^*$-algebra with involution $D_{T_1,T_2}^*=D_{T_2,T_1}$. Let $V$ denotes the closure of $V^0$ inside $B(X)$.
Is $V$ isomorphic to some well known $C^*$-algebra?
I was trying to classify it with $B(H)$ but was unable to do it. Any hints?