Maps preserving the local spectral radius zero of generalized product of operators

On the pseudospectrum preservers

“…The usual product T S, the triple product T ST and the Jordan product T S + ST are a particular cases of this products. Note that maps preserving certain spectral quantities of the generalized product has been considered by many authors; see for instance [1,3].…”

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

“…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.…”

“…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 :…”

Maps preserving fixed points of generalized product of operators

“…The problem of characterizing maps on matrices or operators that preserve certain functions, subsets and relations has attracted the attention of many mathematicians in the last decade; for example we can see [1,2,3,4,5,6,7] and their references. In recent years, a great activity has occurred in characterising maps preserving the subspace of fixed points of a matrix or operators.…”

“…The following theorem is the main result of this paper. Its proof use ideas from [1,7]. Theorem 1.1.…”