摘要
Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, 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 Be∞^＊, 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 〈 ｜da｜ 〈 δ}= θ for some 5 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 tno satisfying tn ∈ U for all n ≠ n0）. If {n ： ｜dn｜ 〈 δ} = θ, 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 N satisfying XAn ∈ R（T）.
Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, 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 Be∞^＊, 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 〈 ｜da｜ 〈 δ}= θ for some 5 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 tno satisfying tn ∈ U for all n ≠ n0）. If {n ： ｜dn｜ 〈 δ} = θ, 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 N satisfying XAn ∈ R（T）.
