On maps preserving operators of local spectral radius zero

Preservers of the local spectral radius zero of Jordan product of operators

“…This result has been extended by M.Elhodaibi and A.Jaatit in [17] to maps ϕ assumed to be nonsurjective. For generalized product of operators, Z.Abdelali, A.Achchi and R.Marzouki showed in [1] that a surjective map ϕ on B(X) satisfies :…”

“…They studied additive maps on B(X) preserving the local spectrum at each point x ∈ X and showed that if ϕ : B(X) −→ B(X) is an additive map satisfying σ ϕ(T ) (x) = σ T (x) for all T ∈ B(X) and for all x ∈ X , then ϕ is the identity on B(X). These results opened the way for some authors to consider a more general problem of characterizing additive or linear maps on matrices or operators preserving different local spectral sets and quantities such as the local spectrum, the local spectral radius and the local inner spectral radius; see for instance [1,[4][5][6][7][8][9][10][11][12][13][15][16][17][18][19] and the references therein.…”