On additive maps preserving certain semi-Fredholm subsets |
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Authors: | Mostafa Mbekhta Mourad Oudghiri |
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Affiliation: | 1. UFR de Mathématiques , Université Lille 1 , 59655 Villeneuve Cedex , France Mostafa.Mbekhta@math.univ-lille1.fr;3. Laboratoire LAGA , Faculté des Sciences d'Oujda , 60000 Oujda , Morocco |
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Abstract: | Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ?(X)?→??(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and ?emrl [M. Mbekhta and P. ?emrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n?≥?1, if φ preserves in both directions ? n (X) (resp., 𝒬 n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T)?=?UTV for all T or φ(T)?=?UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces. |
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Keywords: | additive preservers semi-Fredholm operators |
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