Fredholm operators on the space of bounded sequences

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• 作者：Egor A. ALEKHNO ^{alekhno@bsu.by" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fredholm operator ; space ℓ_&infin ; ℓ&infin ; Stone-Čech compactification β
• 刊名：Acta Mathematica Scientia
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：36
• 期：2
• 页码：614-634
• 全文大小：440 K

文摘

Necessary and sufficient conditions are studied that a bounded operator $T_x=\left(x_1^{*}x,x_2^{*}x,\mathrm{\dots }\right)$ on the space ℓ_&infin;${\mathrm{<font\; color="red">\ell </font>}}_{\mathrm{\&<font\; color="red">infin</font>;}}$, where $x_n^{*}\in {\mathrm{<font\; color="red">\ell </font>}}_{\mathrm{\&<font\; color="red">infin</font>;}}^{*}$, is lower or upper semi-Fredholm; in particular, topological properties of the set $\left\{x_1^{*},x_2^{*},\mathrm{\dots }\right\}$ 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 $x_n^{*}=d_n{x}_{t_n}^{*},$ where $d_n\in \mathcal{R}\text{and}{x}_{t_n}^{*}\ge 0$ are extreme points of the unit ball $B_{{\mathrm{<font\; color="red">\ell </font>}}_{\mathrm{\&<font\; color="red">infin</font>;}}^{*}},$ that is, t_n∈βN,$t_n\in \beta \mathcal{N},$ 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} = φ for some Δ is sufficient and if for large n points tn 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$t_n\notin u$ for all n ≠ n₀). If {n : |d_n| < Δ} = φ, then d(T) is equal to the defect Δ{t_n} of {t_n}. It is shown that if d(T) = &infin; and R(T) is closed, then there exists a sequence {A_n} of pairwise disjoint subsets of satisfying χ_An∉R(T)${\chi }_{A_n}\notin R\left(T\right)$.