数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (2): 614-634.doi: 10.1016/S0252-9602(16)30025-X
• 论文 • 上一篇
Egor A. ALEKHNO
Egor A. ALEKHNO
摘要:
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 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 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}=Ø 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}=Ø, 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 χAn∉R(T).
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