Preservers of local spectrum of matrix Jordan triple products

Maps preserving the local spectrum of Jordan product of matrices

“…With these lemmas, the proof of the main results of [4,5] could be enormously simplified. For example, by using Lemma 5.1 one could get rid of Steps 3, 4, 5 and 6 of the proof of the main result of [4] and simplify such a proof.…”

“…For example, by using Lemma 5.1 one could get rid of Steps 3, 4, 5 and 6 of the proof of the main result of [4] and simplify such a proof. Indeed, assume that for some fixed vector x 0 in C n , a map ϕ from M n (C) into itself satisfies σ ϕ(T )ϕ(S)ϕ(T ) (x 0 ) = σ T ST (x 0 ), (T, S ∈ M n (C)) (5.17)…”

“…Mainly, several authors described maps on matrices or operators that preserve local spectrum, local spectral radius, and local inner spectral radius; see for instance [4,5,[7][8][9][10][11][12][13][14][15][17][18][19][20] and the references therein. In [9,10], nonlinear surjective maps on Banach space operators preserving the local spectrum of the product and the triple product of operators have been investigated.…”

“…In [7], the current authors described the form of all surjective maps preserving the local spectrum of Jordan product of operators on a complex Banach space. In [4,5], nonlinear maps on M n (C) preserving the local spectrum of the product and the triple product of operators have been characterized. The aim of this paper is to characterize the form of all maps (not supposed to be surjective or even linear) on M n (C) that preserve the local spectrum of the Jordan product of matrices at a nonzero fixed vector in C n .…”

“…Although the proof of such a promised result uses some arguments which are influenced by ideas from several papers including [16], it requires new ingredients which will be established in Section 3. It is also somehow simpler than the proofs of the main results of [4,5] as will be explained in the last section of this paper. Furthermore, it is different from that of [7] wherein the characterization of rank one nilpotent operators is employed.…”