Product of operators and numerical range preserving maps

Chi-Kwong Li, and Nung-Sing Sze. "Product of operators and numerical range preserving maps." Studia Mathematica 174.2 (2006): 169-182. <http://eudml.org/doc/285089>.

@article{Chi2006,
abstract = {Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A₁,...,A_k ∈ V$ by $A₁* ⋯ * A_k = A_\{i₁\} ⋯ A_\{iₘ\}$. This includes the usual product $A₁* ⋯ * A_k = A₁ ⋯ A_k$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying $μ^\{m\} = 1$ such that ϕ: V → V has the form A ↦ μU*AU or $A ↦ μU*A^\{t\}U$, then ϕ is surjective and satisfies $W(A₁ * ⋯ *A_k) = W(ϕ(A₁)* ⋯ *ϕ(A_k))$ for all $A₁, ..., A_k ∈ V$. It is shown that the converse is true under the assumption that one of the terms in (i₁, ..., iₘ) is different from all other terms. In the finite-dimensional case, the converse can be proved without the surjectivity assumption on ϕ. An example is given to show that the assumption on (i₁, ..., iₘ) is necessary.},
author = {Chi-Kwong Li, Nung-Sing Sze},
journal = {Studia Mathematica},
keywords = {numerical range; Jordan triple product},
language = {eng},
number = {2},
pages = {169-182},
title = {Product of operators and numerical range preserving maps},
url = {http://eudml.org/doc/285089},
volume = {174},
year = {2006},
}

TY - JOUR
AU - Chi-Kwong Li
AU - Nung-Sing Sze
TI - Product of operators and numerical range preserving maps
JO - Studia Mathematica
PY - 2006
VL - 174
IS - 2
SP - 169
EP - 182
AB - Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A₁,...,A_k ∈ V$ by $A₁* ⋯ * A_k = A_{i₁} ⋯ A_{iₘ}$. This includes the usual product $A₁* ⋯ * A_k = A₁ ⋯ A_k$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying $μ^{m} = 1$ such that ϕ: V → V has the form A ↦ μU*AU or $A ↦ μU*A^{t}U$, then ϕ is surjective and satisfies $W(A₁ * ⋯ *A_k) = W(ϕ(A₁)* ⋯ *ϕ(A_k))$ for all $A₁, ..., A_k ∈ V$. It is shown that the converse is true under the assumption that one of the terms in (i₁, ..., iₘ) is different from all other terms. In the finite-dimensional case, the converse can be proved without the surjectivity assumption on ϕ. An example is given to show that the assumption on (i₁, ..., iₘ) is necessary.
LA - eng
KW - numerical range; Jordan triple product
UR - http://eudml.org/doc/285089
ER -