Holub proved that Fredholm operators are stable under compact perturbations. I am interested in a slight refinement of this theorem.
Suppose we have two operators $T_1$ and $T_2$ acting on a primary Banach space $E$. Assume that the range $(T_1+T_2)(E)$ is isomorphic to $E$ and complemented in $E$. Is it true that there exists $i\in \{1,2\}$ such that the range $T_i(E)$ is isomorphic to $E$ and complemented in $E$?