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

100%
Li C.-K., Poon Y.-T.
Studia Mathematica
|
2009
|
tom 194
|
nr 1
91-104
EN
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 $cl(W(A₁, ..., A_{i+1}, A_{i} + F, A_{i+1}, ..., Aₘ))$, 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. Although cl(W(A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d ∉ cl(W(A)), there is a linear functional f such that f(d) > sup{f(a): a ∈ cl(W(Ã))}, where à is obtained from A by perturbing one of the components $A_{i}$ by a finite rank self-adjoint operator. Other results on W(A) and $W_{e}(A)$ extending those on a single operator are obtained.
2
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Product of operators and numerical range preserving maps

100%
Li C.-K., Sze N.-S.
Studia Mathematica
|
2006
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tom 174
|
nr 2
169-182
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.
3
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Jordan isomorphisms and maps preserving spectra of certain operator products

81%
Hou J., Li C.-K., Wong N.-C.
Studia Mathematica
|
2008
|
tom 184
|
nr 1
31-47
EN
Let 𝓐₁, 𝓐₂ 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 $𝓐_{i}$, which covers the usual product $T₁ ∗ ⋯ ∗ T_{k} = T₁ ⋯ T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: 𝓐₁ → 𝓐₂ 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.
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