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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.

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    $\begingroup$ 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. $\endgroup$ – fedja May 19 at 4:41

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