Let $X$ be a Topological Vector (or Fréchet) Space and $T:X\to X$ be continuous. Is there a right inverse for $T+\lambda I, \lambda>1$?
For example, let $H(\Bbb C)$ be the set of entire fonctions and let $D:H(\Bbb C)\to H(\Bbb C)$ be given by $D(f)=f'$. Is there a right inverse for $D+\lambda I, \lambda>1$?
If not, is there some kind of "approximate right inverse" with good properties that could be defined for this last example?