Let $T\colon X\to X$ be an upper-semi Fredholm operator acting on a $B$-space $X$ (the range of $T$ is closed and kernel is finite-dimensional) with complemented range. Suppose $S\colon X\to X$ is bounded below. Does it follow that $T+S$ has complemented range? If we have assumed that $S$ is compact, then the answer would be 'yes'.



Perturbing upper-semi Fredholm operators
In fact, *any* bounded operator $L$ on $X$ may be written as a sum of two invertible operators $S$ and $T$, hence in particular both Fredholm and bounded below. We may take $S:=\lambda I$ and $T:=L-\lambda I$, with $\lambda > \|L\|$. 


