The perturbation classes problem in Fredholm theory. (Q1874451)

Let \(X,\) \(Y\) be complex Banach spaces, \(\Phi=\Phi (X)\) be the space of Fredholm operators on \(X,\) \(I_X\) be the identity operator on \(X,\) \(J_M\) be the inclusion of a closed sumspace \(M\) of \(X\) in \(X\) and \(Q_M\) be the quotient map onto \(X/M.\) An operator \(T\in {\mathcal L}(X,Y)\) is inessential if \(I_X-ST\in \Phi (X),\) for every \(S\in {\mathcal L}(Y,X);\) it is strictly singular if no restriction \(TJ_M\) of \(T\) to a closed infinite-dimensional subspace \(M\) of \(X\) is an isomorphism; and it is strictly cosingular if there is no closed infinite-codimensional subspace \(N\) of \(Y\) such that \(Q_N T\) is surjective. It is well-known that the perturbation class of \(\Phi \) is the set of inessential operators. The main purpose of the paper under review is to give a negative answer to the important perturbation class problems: the perturbation class of the upper semi-Fredholm operators contains properly the strictly singular operators, and the perturbation class of the lower semi-Fredholm operators contains properly the strictly cosingular operators.