Quantities characterizing semi-Fredholm operators and perturbation radii (Q408290)

There are several different quantities \(q\) which characterize the class \(\Phi_+\) of the upper semi-Fredholm operators as follows: \(q(T)>0 \iff T\in \Phi_+\). Already in [Stud. Math. 114, No. 1, 13--27 (1995; Zbl 0830.47008)], the authors have determined many relations among these quantities. For instance, they have observed that the quantities are classified in three equivalence classes, and that all of them are smaller than the perturbation radius of \(T\), denoted by \(d_+(T)\), which is defined as the distance of \(T\) to the complement of \(\Phi_{+}\). However, a few questions have remained open. For instance, it was not known if the quantity \(\text{in}_+(T)=\inf \| T|_M\|\), where \(T|_M\) denotes the restriction of \(T\) to a subspaces \(M\) and the infimum in the definition is taken over all closed infinite-dimensional subspaces, is equivalent to \(d_+\).NEWLINENEWLINEIn this paper, it is shown that there exist two Banach spaces and a sequence of operators between them such that \(\text{in}_+(T_n)/d_+(T_n) \to 0\) as \(n\to \infty\). It follows from this that \(\text{in}_+\) and \(d_+\) are not equivalent in general. However, as the equivalence of these two quantities is more interesting, the paper ends with examples of pairs of Banach spaces such that \(\text{in}_+\) and \(d_+\) are equivalent.