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Fredholm operators on the space of bounded sequences

Research paper by Egor A. ALEKHNO

Indexed on: 17 Mar '16Published on: 02 Feb '16Published in: Acta Mathematica Scientia



Abstract

Necessary and sufficient conditions are studied that a bounded operator <img height="20" border="0" style="vertical-align:bottom" width="121" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S025296021630025X-si1.gif">Tx=(x1*x,x2*x,…) on the space ℓ∞ℓ∞, where <img height="20" border="0" style="vertical-align:bottom" width="59" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S025296021630025X-si3.gif">xn*∈ℓ∞*, is lower or upper semi-Fredholm; in particular, topological properties of the set <img height="20" border="0" style="vertical-align:bottom" width="76" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S025296021630025X-si4.gif">{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 <img height="24" border="0" style="vertical-align:bottom" width="76" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S025296021630025X-si5.gif">xn*=dnxtn*, where <img height="24" border="0" style="vertical-align:bottom" width="131" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S025296021630025X-si6.gif">dn∈R and xtn*≥0 are extreme points of the unit ball <img height="18" border="0" style="vertical-align:bottom" width="32" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S025296021630025X-si7.gif">Bℓ∞*, that is, tn∈βN,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∉utn∉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 <img height="9" border="0" style="vertical-align:bottom" width="9" alt="Full-size image (8 K)" title="Full-size image (8 K)" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S025296021630025X-fx1.jpg"> satisfying χAn∉R(T)χAn∉R(T).