Sum of a closed range operator and a non closed range operator

Question

Let $T, T'\in B(H)$ such that $T$ is closed range and $T'$ is not. Is $T+T'$ is closed range or not? Is there any assumption on $T$ or $T'$ that $T+T'$ is closed range?

Answer 1

Either case can happen. For example, if $T'$ does not have a closed range, then $0+T'$ does not have a closed range. If $T'$ does not have a closed range, and $\lambda$ is not in the spectrum of $T'$ then $\lambda I+T'$ has closed range because it is surjective.