On a algebraic characterization of the generalized Fredholm operators

We will give an algebraic characterization of generalized Fredholm operators in terms of projections by means of Schmoeger (Demonstr Math XXXII(3):595–604, 1999, Therorem 1.1). More precisely, for T ∈ B ( X ) which is the Banach algebra of all bounded linear operators on a Banach space X , we shall...

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Bibliographic Details
Published in:Rendiconti del Circolo matematico di Palermo 2024-11, Vol.73 (7), p.2635-2642
Main Authors: Hadder, Youness, El Amrani, Abdelkhalek, Bourouaha, Imad
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Banach spaces
Geometry
Linear operators
Mathematics
Mathematics and Statistics
Operators (mathematics)
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Summary:We will give an algebraic characterization of generalized Fredholm operators in terms of projections by means of Schmoeger (Demonstr Math XXXII(3):595–604, 1999, Therorem 1.1). More precisely, for T ∈ B ( X ) which is the Banach algebra of all bounded linear operators on a Banach space X , we shall prove the following fact: T is a generalized Fredholm operator if and only if there exists a projection P which commutes with T such that, TP is a Fredholm element in P B ( X ) P with jump j P B ( X ) P ( T P ) = 0 and ( I d X - P ) T is a nilpotent element in s o c ( ( I d X - P ) B ( X ) ( I d X - P ) ) . This characterization can open the way for us to extend the above Theorem to the more general context like that of a semisimple complex Banach algebras.
ISSN:0009-725X
1973-4409
DOI:10.1007/s12215-024-01059-7