Jordan isomorphisms and maps preserving spectra of certain operator products

Jinchuan Hou, Chi-Kwong Li, and Ngai-Ching Wong. "Jordan isomorphisms and maps preserving spectra of certain operator products." Studia Mathematica 184.1 (2008): 31-47. <http://eudml.org/doc/284893>.

@article{JinchuanHou2008,
abstract = {Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁ ∗ ⋯ ∗ T_\{k\}$ on elements in $_\{i\}$, which covers the usual product $T₁ ∗ ⋯ ∗ T_\{k\} = T₁ ⋯ T_\{k\}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $σ(Φ(A₁) ∗ ⋯ ∗ Φ(A_\{k\})) = σ(A₁ ∗ ⋯ ∗ A_\{k\})$ whenever any one of $A_\{i\}$’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.},
author = {Jinchuan Hou, Chi-Kwong Li, Ngai-Ching Wong},
journal = {Studia Mathematica},
keywords = {standard operator algebra; spectral functions; Jordan triple products of operators; skew products of operators; nonlinear preserver problems},
language = {eng},
number = {1},
pages = {31-47},
title = {Jordan isomorphisms and maps preserving spectra of certain operator products},
url = {http://eudml.org/doc/284893},
volume = {184},
year = {2008},
}

TY - JOUR
AU - Jinchuan Hou
AU - Chi-Kwong Li
AU - Ngai-Ching Wong
TI - Jordan isomorphisms and maps preserving spectra of certain operator products
JO - Studia Mathematica
PY - 2008
VL - 184
IS - 1
SP - 31
EP - 47
AB - Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁ ∗ ⋯ ∗ T_{k}$ on elements in $_{i}$, which covers the usual product $T₁ ∗ ⋯ ∗ T_{k} = T₁ ⋯ T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $σ(Φ(A₁) ∗ ⋯ ∗ Φ(A_{k})) = σ(A₁ ∗ ⋯ ∗ A_{k})$ whenever any one of $A_{i}$’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.
LA - eng
KW - standard operator algebra; spectral functions; Jordan triple products of operators; skew products of operators; nonlinear preserver problems
UR - http://eudml.org/doc/284893
ER -