Suppose $H=H_1\oplus H_2\oplus...$ is an orthogonal decomposition of a Hilbert space $H$. Let $T:\mathrm{dom}(T)\subseteq H\to H$ a densely defined linear operator (it could be not bounded). For each $n$, let $P_n$ the orthogonal projection from $H$ onto $H_n$.
Is it true that $T=\sum_{n,k}P_nTP_k$?
I found this equality in a paper, but I think that it is not true.