# Decomposition of an unbounded operator

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.

• Some additional conditions are certainly due. As written, we may have all sorts of trouble like $P_kH\cap \rm{dom}(T)=\{0\}$ for all $k$, etc. – fedja May 19 at 4:41