# Linear maps preserving the generalized spectrum.

Extracta Mathematicae (2007)

• Volume: 22, Issue: 1, page 45-54
• ISSN: 0213-8743

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## Abstract

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Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let σg(T) denote the generalized spectrum of T. In this paper, we prove that if φ: B(H) → B(H) is a surjective linear map, then φ preserves the generalized spectrum (i.e. σg(φ(T)) = σg(T) for every T ∈ B(H)) if and only if there is A ∈ B(H) invertible such that either φ(T) = ATA-1 for every T ∈ B(H), or φ(T) = ATtrA-1 for every T ∈ B(H). Also, we prove that γ(φ(T)) = γ(T) for every T ∈ B(H) if and only if there is U ∈ B(H) unitary such that either φ(T) = UTU* for every T ∈ B(H) or φ(T) = UTtrU* for every T ∈ B(H). Here γ(T) is the reduced minimum modulus of T.

## How to cite

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Mbekhta, Mostafa. "Linear maps preserving the generalized spectrum.." Extracta Mathematicae 22.1 (2007): 45-54. <http://eudml.org/doc/41869>.

@article{Mbekhta2007,
abstract = {Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let σg(T) denote the generalized spectrum of T. In this paper, we prove that if φ: B(H) → B(H) is a surjective linear map, then φ preserves the generalized spectrum (i.e. σg(φ(T)) = σg(T) for every T ∈ B(H)) if and only if there is A ∈ B(H) invertible such that either φ(T) = ATA-1 for every T ∈ B(H), or φ(T) = ATtrA-1 for every T ∈ B(H). Also, we prove that γ(φ(T)) = γ(T) for every T ∈ B(H) if and only if there is U ∈ B(H) unitary such that either φ(T) = UTU* for every T ∈ B(H) or φ(T) = UTtrU* for every T ∈ B(H). Here γ(T) is the reduced minimum modulus of T.},
author = {Mbekhta, Mostafa},
journal = {Extracta Mathematicae},
keywords = {reduced minimum modulus; generalized spectrum; Jordan isomorphism},
language = {eng},
number = {1},
pages = {45-54},
title = {Linear maps preserving the generalized spectrum.},
url = {http://eudml.org/doc/41869},
volume = {22},
year = {2007},
}

TY - JOUR
AU - Mbekhta, Mostafa
TI - Linear maps preserving the generalized spectrum.
JO - Extracta Mathematicae
PY - 2007
VL - 22
IS - 1
SP - 45
EP - 54
AB - Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let σg(T) denote the generalized spectrum of T. In this paper, we prove that if φ: B(H) → B(H) is a surjective linear map, then φ preserves the generalized spectrum (i.e. σg(φ(T)) = σg(T) for every T ∈ B(H)) if and only if there is A ∈ B(H) invertible such that either φ(T) = ATA-1 for every T ∈ B(H), or φ(T) = ATtrA-1 for every T ∈ B(H). Also, we prove that γ(φ(T)) = γ(T) for every T ∈ B(H) if and only if there is U ∈ B(H) unitary such that either φ(T) = UTU* for every T ∈ B(H) or φ(T) = UTtrU* for every T ∈ B(H). Here γ(T) is the reduced minimum modulus of T.
LA - eng
KW - reduced minimum modulus; generalized spectrum; Jordan isomorphism
UR - http://eudml.org/doc/41869
ER -

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