Fredholm operators on the space of bounded sequences

Abstract

Necessary and sufficient conditions are studied that a bounded operator Tx=(x1*x,x2*x,…) on the space ℓ∞, where xn*∈ℓ∞*, is lower or upper semi-Fredholm; in particular, topological properties of the set {x1*,x2*,…} are investigated. Various estimates of the defect d(T) = codim R(T), where R(T) is the range of T, are given. The case of xn*=dnxtn*, where dn∈R and xtn*≥0 are extreme points of the unit ball Bℓ∞*, that is, tn∈βN, is considered. In terms of the sequence {tn}, the conditions of the closedness of the range R(T) are given and the value d(T) is calculated. For example, the condition {n : 0 < |dn| <Δ} = φ for some Δ is sufficient and if for large n points tn are isolated elements of the sequence {tn}, then it is also necessary for the closedness of R(T) (tn0 is isolated if there is a neighborhood u of tn0 satisfying tn∉u for all n ≠ n0). If {n : |dn| <Δ} = φ, then d(T) is equal to the defect Δ{tn} of {tn}. It is shown that if d(T) = ∞ and R(T) is closed, then there exists a sequence {An} of pairwise disjoint subsets of ▪ satisfying χAn∉R(T).