Product of operators and numerical range preserving maps

• Autorzy: Chi-Kwong Li , Nung-Sing Sze
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47B49: Transformers, preservers (operators on spaces of operators)
• 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
• 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)
• 15A04: Linear transformations, semilinear transformations
• 47A12: Numerical range, numerical radius
• 15A18: Eigenvalues, singular values, and eigenvectors

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 174
• Numer: 2
• Strony: 169-182

Daty

wydano

2006

Twórcy

autor

Chi-Kwong Li

• Department of Mathematics, College of William and Mary, Williamsburg, VA 23185, U.S.A.

autor

Nung-Sing Sze

• Department of Mathematics, University of Hong Kong, Hong Kong

Identyfikatory

DOI

10.4064/sm174-2-4