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Product of operators and numerical range preserving maps

Chi-Kwong Li;  Nung-Sing Sze — 2006

Studia Mathematica

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 A ₁ , . . . , A k ∈ V by A ₁ * ⋯ * A k = A i ₁ ⋯ A i ₘ . This includes the usual product A ₁ * ⋯ * A k = A ₁ ⋯ 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...

The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li;  Yiu-Tung Poon — 2009

Studia Mathematica

Let W(A) and W e ( A ) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that W e ( A ) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, W e ( A ) can be obtained as the intersection of all sets of the form c l ( W ( A ₁ , . . . , A i + 1 , A i + F , A i + 1 , . . . , A ₘ ) ) , where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in W e ( A ) as star centers....

Jordan isomorphisms and maps preserving spectra of certain operator products

Jinchuan Hou;  Chi-Kwong Li;  Ngai-Ching Wong — 2008

Studia Mathematica

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 T ₁ ∗ ⋯ ∗ T k on elements in i , which covers the usual product T ₁ ∗ ⋯ ∗ T k = T ₁ ⋯ T k and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying σ ( Φ ( A ₁ ) ∗ ⋯ ∗ Φ ( A k ) ) = σ ( A ₁ ∗ ⋯ ∗ 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...

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