A generalization of a classic theorem in the perturbation theory for linear operators (Q1961034)

Let \(T\) be a bounded linear operator of a Banach space \(X\) into another \(Y\) such that the null space \(N(T)\) is a complementary subspace of \(X\) and the range \(R(T)\) a complementary subspace of \(Y\). This note considers the equation \(Tx= y\) for given \(y\) and its perturbed one \((T+\delta T)\widetilde x= y+\delta y\), to estimate the norm of the difference between the solutions \(x\) and \(\widetilde x\). The argument used is essentially based on the Neumann lemma to the effect that for a bounded linear operator \(S\) on a Banach space, \(I+S\) has a bounded inverse if \(\|S\|< 1\).