Open Access

# 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.

### Article Information

 Title Linear maps preserving the index of operators Source Methods Funct. Anal. Topology, Vol. 23 (2017), no. 3, 277-284 MathSciNet MR3707522 Milestones Received 20/12/2016; Revised 30/03/2017 Copyright The 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

### 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},
MRNUMBER = {MR3707522},
URL = {http://mfat.imath.kiev.ua/article/?id=989},
}

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