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Linear maps preserving the index of operators


Abstract

Let $\mathsf{H}$ be an infinite-dimensional separable complex Hilbert space and $\mathcal{B}(\mathsf{H})$ the algebra of all bounded linear operators on $\mathsf{H}.$ In this paper, we prove that if a surjective linear map $ \phi : \mathcal{B}(\mathsf{H}) \longrightarrow \mathcal{B}(\mathsf{H})$ preserves the index of operators, then $\phi$ preserves compact operators in both directions and the induced map $ \varphi : \mathcal{C}( \mathsf{H}) \longrightarrow \mathcal{C}(\mathsf{H}),$ determined by $\varphi(\pi(T)) = \pi( \phi(T)) $ for all $T \in \mathcal{B}(\mathsf{H}),$ is a continuous automorphism multiplied by an invertible element in $\mathcal{C}( \mathsf{H}).$

Key words: Linear preserver problems, index of operator, semi-Fredholm operator.


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Article Information

TitleLinear maps preserving the index of operators
SourceMethods Funct. Anal. Topology, Vol. 23 (2017), no. 3, 277-284
MilestonesReceived 20/12/2016; Revised 30/03/2017
CopyrightThe Author(s) 2017 (CC BY-SA)

Authors Information

Sayda Ragoubi
Universite de Monastir, Institut pr ´eparatoire aux´ etudes d’ing´ enieurs de Monastir, Avenue Ibn Eljazzar, 5019 Monastir, Tunisia 


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Citation Example

Sayda Ragoubi, Linear maps preserving the index of operators, Methods Funct. Anal. Topology 23 (2017), no. 3, 277-284.


BibTex

@article {MFAT989,
    AUTHOR = {Ragoubi, Sayda},
     TITLE = {Linear maps preserving the index of operators},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {23},
      YEAR = {2017},
    NUMBER = {3},
     PAGES = {277-284},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=989},
}


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