Given a Banach space $X$ which can be written a direct sum of two subspaces $Y\oplus Z$ and the $u\in B(X), w\in B(Y), v\in B(Z)$ and $u=w\oplus v$. where $B(\cdot)$ denotes the space of bounded operators on the space.
So is it true that $u$ is compact iff both $v$ and $w$ are? If not, what would be the counter-example?
The $\Rightarrow$ is indeed true, as seen in Direct sum of compact operators is compact. Nonetheless, the reverse direction requires that every compact operators to be the norm-limit of finite-rank operators, which is not the case in general Banach spaces.
P.S. If $X, Y, Z$ are all Hilbert spaces, this is indeed true. The same link above addresses the question.