Solve commutator relation $[Q,d]=-[P,d]$ for $Q$ on chain complexes with scalar product

Suppose we are given chain sequences

$\dots \rightarrow C_k \rightarrow C_{k+1} \rightarrow \dots$

and

$\dots \rightarrow D_k \rightarrow D_{k+1} \rightarrow \dots$

of finite-dimensional vector spaces with scalar product $\langle \cdot,\cdot \rangle$. The differentials are denoted by $d_k$. The spaces $D_k$ are subspaces of the $C_k$, respectively, and $P_k$ is the orthogonal projection from each $C_k$ onto $D_k$.

In general $P$ and $d$ do not commute, whence $P$ is not a cochain map. But any commuting cochain projection $S$ from $C_k$ onto $D_k$, that does leave $D_k \subset C_k$ invariant, can be written as $S = P + Q$, where $Q$ is a linear operator that maps $C_k$ to $D_k$ such that

• It vanishes on $D_k \subset C_k$
• We have $[Q,d] = - [P,d]$

where $[\cdot,\cdot]$ is the commutator of linear mappings. In other words, $Q$ is a "corrector" which cancels out the failure of $P$ to commute.

However, I do not know under what conditions such a map exists. As the endomorphism of $\oplus_k C_k$ are a Lie algebra, one might suspect this is a basic question in algebra. Unfortunately, my knowledge in this area is rather surficial. I would be nice if someone could give me a hint where to learn about the solvability of these equations.

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