Acta mathematica scientia, Series B >
FREDHOLM OPERATORS ON THE SPACE OF BOUNDED SEQUENCES
Received date: 2014-03-10
Revised date: 2015-07-08
Online published: 2016-04-25
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).
Key words： Fredholm operator; space l∞; Stone-?ech compactification βN
Egor A. ALEKHNO . FREDHOLM OPERATORS ON THE SPACE OF BOUNDED SEQUENCES[J]. Acta mathematica scientia, Series B, 2016 , 36(2) : 614 -634 . DOI: 10.1016/S0252-9602(16)30025-X
[1] Abramovich Y A, Aliprantis C D. An invitation to operator theory. Graduate Studies in Mathematics. Vol 50. Providence, RI, USA:American Mathematical Society, 2002
[2] Aiena P. Fredholm and local spectral theory, with applications to multipliers. Dordrecht, Netherlands:Kluwer Academic Publishers, 2004
[3] Alekhno E A. The lower Weyl spectrum of a positive operator. Integr Equ Oper Theory, 2010, 67(3):301-326
[4] Alekhno E A. Superposition operator on the space of sequences almost converging to zero. Cent Eur J Math, 2012, 10(2):619-645
[5] Aliprantis C D, Burkinshaw O. Positive Operators. Orlando, Fla, USA:Academic Press, 1985
[6] Bourbaki N. Topological Vector Spaces. Berlin, Germany:Springer, 1987
[7] Caradus S R, Pfaffenberger W E, Yood B. Calkin Algebras and Algebras of Operators on Banach Spaces. New York, NY, USA:Marcel Dekker, 1974
[8] Drewnowski L. Some characterizations of semi-Fredholm operators. Rocz PTM, 1984, 24(2):215-218
[9] Gillman L, Jerison M. Rings of Continuous Functions. Princeton, NJ, USA:Van Nostrand, 1960
[10] Krasnosel'skij M A, Lifshits Je A, Sobolev A V. Positive linear systems:The method of positive operators. Berlin, Germany:Heldermann, 1989
[11] Schaefer H H. Banach Lattices and Positive Operators. Berlin, Germany:Springer, 1974
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