Surjective maps preserving quasi-nilpotence of products of operators
Abstract
Let X be an infinite-dimensional complex Banach space and let ℬ(X) the full algebra of all bounded linear operator on X. In this paper, we give a complete characterization of surjective maps ϕ:ℬ(X)→ℬ(X) satisfying r(AB)=0 Longleftrightarrow r(ϕ(A)ϕ(B))=0 (A,B∈ℬ(X)). Without assuming additivity, injectivity, or continuity, we show that there exists a function γ: B(X)↦ C ∖{0} such that the map ϕ takes one of the following forms: 1. there exists a bounded linear or conjugate-linear, invertible operator S:X→ X such that : \label{eq:direct} ϕ(A)=γ(A)SAS^{-1},\qquad \text{ for all } \quad A\in B(X), 2. the space X is reflexive and there exists a bounded linear or conjugate-linear, invertible operator S:X*→ X such that \label{eq:dual} ϕ(A)=γ(A)SA^*S^{-1},\qquad \text{ for all } \quad A\in B(X).
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BibTeX
@misc{azna2026surjective,
title = {{Surjective maps preserving quasi-nilpotence of products of operators}},
author = {Zakariae Azna and Fadil Chabbabi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21840},
eprint = {2610.21840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21840},
}
RIS
TY - UNPB
TI - Surjective maps preserving quasi-nilpotence of products of operators
AU - Zakariae Azna
AU - Fadil Chabbabi
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21840
UR - https://arxiv.org/abs/2610.21840
AB - Let X be an infinite-dimensional complex Banach space and let ℬ(X) the full algebra of all bounded linear operator on X. In this paper, we give a complete characterization of surjective maps ϕ:ℬ(X)→ℬ(X) satisfying r(AB)=0 Longleftrightarrow r(ϕ(A)ϕ(B))=0 (A,B∈ℬ(X)). Without assuming additivity, injectivity, or continuity, we show that there exists a function γ: B(X)↦ C ∖{0} such that the map ϕ takes one of the following forms: 1. there exists a bounded linear or conjugate-linear, invertible operator S:X→ X such that : \label{eq:direct} ϕ(A)=γ(A)SAS^{-1},\qquad \text{ for all } \quad A\in B(X), 2. the space X is reflexive and there exists a bounded linear or conjugate-linear, invertible operator S:X^*→ X such that \label{eq:dual} ϕ(A)=γ(A)SA^*S^{-1},\qquad \text{ for all } \quad A\in B(X).
ER -