Lie product and local spectral subspace preservers
Keywords:
Local spectral subspace, Lie product, Preserver problemAbstract
Consider a complex Banach space, denoted by X with dim X≥4, and B(X) represents the algebra of all bounded linear operators on X. For afixed complex scalar λ0, define XS({λ0}) as the local spectral subspace ofan operator S ∈ B(X) associated with {λ0}. We provide a characterizationof all mapsφonB(X)whose range includes operators of rank at mostfour, and that satisfy
Xφ(S)φ(T)−φ(T)φ(S)({λ0}) =XST−T S({λ0})
for all S,T∈B(X).
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Copyright (c) 2026 Somaya Saber, Mhamed Elhodaibi , Soufiane Elouazzani
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