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Joint subnormality of a family of composition operators on L²-space is characterized by means of positive definiteness of appropriate Radon-Nikodym derivatives. Next, simplified positive definiteness conditions guaranteeing joint subnormality of a C₀-semigroup of composition operators are supplied. Finally, the Radon-Nikodym derivatives associated to a jointly subnormal C₀-semigroup of composition operators are shown to be the Laplace transforms of probability measures (modulo a C₀-group of scalars)...

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 $T ₁ * \dots * T_{k}$ on elements in $_{i}$ , which covers the usual product $T ₁ * \dots * T_{k} = T ₁ \dots T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $σ (Φ (A ₁) * \dots * Φ (A_{k})) = σ (A ₁ * \dots * A_{k})$ 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...

Julia's Lemma for Operators.

Ky Fan (1979)

Mathematische Annalen

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Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras [Book]

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