On the perturbation of unbounded linear operators with topologically complemented ranges (Q749838)

Let X and Y be normed spaces and let \(T: D(T)\subset X\to Y\) be a linear transformation having a finite codimensional restriction with a continuous inverse (for example, if T is bounded below). Conditions are obtained under which \(range T+S\) \((R(T+S))\), resp. \(\overline{R(T+S)}\), is topological complemented, if R(T) (or \(\overline{R(T)})\) is topologically complemented in Y (operators S belong to the class of precompact operators, or to some wider class).