Projection on sum of Banach spaces

Question

If we have projections $P$ and $Q$ on Banach spaces $X$ and $Y$, respectively. It is easy to see that the operator matrix $$ S=\begin{bmatrix} P & 0 \\ 0 & Q \end{bmatrix} $$ determines a projection on $X\oplus Y$ such that $R(S)= R(P) \oplus R(Q)$ and $N(S)=N(P)\oplus N(Q)$. My question is this. If I have a projection $S$ on $X\oplus Y$, what would be the proper representation of this projection in the form of a $2\times 2$ operator matrix ? Would it be a diagonal operator matrix ?

Answer 1

No. Let $x \in \mathbb{R}^2$ be a vector of norm $1$, but not aligned with the coordinate axes. Then, $x \, x^\top$ is a projection which is not block-diagonal.