Nonlinear maps preserving the local spectral subspace

Benbouziane, Hassane; Kettani, Mustapha Ech-Chérif El; Herrou, Imane

doi:10.1080/03081087.2017.1409693

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Cited by 5 publications

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“…More precisely, they proved that the only additive map ϕ on B(X) for which X ϕ(T ) ({λ}) = X T ({λ}) for all T ∈ B(X) and λ ∈ C, is the identity. In [4], Benbouziane et al characterized the forms of surjective weakly continuous maps ϕ from B(X) into B(X) which satisfy…”

Section: Introductionmentioning

confidence: 99%

Maps preserving the local spectral subspace of skew-product of operators

Parvinianzadeh¹

2022

Preprint

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Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H. For T ∈ B(H) and λ ∈ C, let HT ({λ}) denotes the local spectral subspace of T associated with {λ}. We prove that if ϕ : B(H) → B(H) be an additive map such that its range contains all operators of rank at most two and satisfiesfor all T, S ∈ B(H) and λ ∈ C, then there exist a unitary operator V in B(H) and a nonzero scalar µ such that ϕ(T ) = µT V * for all T ∈ B(H). We also show if ϕ1 and ϕ2 be additive maps from B(H) into B(H) such that their ranges contain all operators of rank at most two and satisfies) * ({λ}) = HT S * ({λ}) for all T, S ∈ B(H) and λ ∈ C. Then ϕ2(I) * is invertible, and ϕ1(T ) = T (ϕ2(I) * ) −1 and ϕ2(T ) = ϕ2(I) * T for all T ∈ B(H).

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Section: Introductionmentioning

confidence: 99%