Let $T\in B(H)$ and $H=H_1\oplus H_2$, then $T$ can be expressed by the following matrix:
\begin{pmatrix}A & B\\C & D\end{pmatrix} , where $A\in B(H_1), B\in B(H_2,H_1), C\in B(H_1,H_2),D\in B(H_2)$.
How to represent the operator $T|_{H_1}$ which is restricted to the closed subspace $H_1$ in terms of the matrix? $H_1$ may not be an invariant subspcae of $T$.