The operator matrix on Hilbert space Let $H$ be a Hilbert space and $P$ be the projection operator, then $H= P(H)\oplus (1-P)(H)$. Hence, for each $T\in B(H)$, we have 
$$T=\left(\begin{array}{ccc}
    PTP & PT(1-P) \\
    (1-P)TP & (1-P)T(1-P) \\
  \end{array}\right)$$
I hope to know why we can decompose
an operator $T$ into such a form of matrix?

It is just a fancy (and useful) way of writing the trivial equality
$$
T=PTP+(I-P)TP+PT(I-P)+(I-P)T(I-P).
$$
It is useful because both addition and multiplication behave like the corresponding operations for matrices,  so you can actually treat $T$ as a $2\times2$ matrix (without forgetting that the entries are operators). 
