关键词: Fredholm operator, space l_∞, Stone-?ech compactification βN

Abstract:

Necessary and sufficient conditions are studied that a bounded operator T_x=(x₁^*x, x₂^*x, …) on the space l_∞, where x_n^*∈l_∞^*, is lower or upper semi-Fredholm; in partic-ular, 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 x_n^*=d_nx_tn^*, where d_n∈R and x_tn^*≥0 are extreme points of the unit ball B_l∞^*, that is, tn ∈βN, is considered. In terms of the sequence {t_n}, 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<|d_n|<δ}=Ø for some δ is sufficient and if for large n points t_n are isolated elements of the sequence {t_n}, then it is also necessary for the closedness of R(T) (t_n0 is isolated if there is a neighborhood U of t_n0 satisfying t_n∉U for all n≠n₀). If {n:|d_n|<δ}=Ø, then d(T) is equal to the defect δ{t_n} of {t_n}. It is shown that if d(T)=∞ and R(T) is closed, then there exists a sequence {A_n} of pairwise disjoint subsets of N satisfying χ_An∉R(T).

Key words: Fredholm operator, space l_∞, Stone-?ech compactification βN