A question about essential ideal 
Let $I$ be a nonunital C*-algebra and $I\subset B(H)$ be any nondegenerate representation and define
  $$M(I)=\{T\in B(H): Tx\in I~and ~xT\in I,  ~for ~all~ x\in I\}.$$
  Then,  how to prove $I$ sits in $M(I)$ as an essential ideal?

Definition An ideal $I\triangleleft E$ is essential if it has "no orthogonal complement" - i.e.,
$$I^{\bot}:=\{e\in E: ex=xe=0,~ for ~all~x\in I\}=\{0\}.$$

First of all, see this question. 
If $Tx=xT=0$ for every $x\in I$ then for every $h\in H$ we have $Txh=0$.
Now the set $\{xh,\ x\in I, h\in H\}$ is dense in $H$, since the representation is non-degenerate by the question above. 
Since $\ker(T)$ is closed in $H$ and contains $\{xh,\ x\in I, h\in H\}$, which is dense in $H$ then $\ker(T)=H$ and $T=0$.
