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## Studia Mathematica

2006 | 174 | 2 | 169-182
Tytuł artykułu

### Product of operators and numerical range preserving maps

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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.
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Tom
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169-182
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wydano
2006
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autor
• Department of Mathematics, College of William and Mary, Williamsburg, VA 23185, U.S.A.
autor
• Department of Mathematics, University of Hong Kong, Hong Kong
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