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Jacobi matrices on trees

Agnieszka M. Kazun, Ryszard Szwarc (2010)

Colloquium Mathematicae

Symmetric Jacobi matrices on one sided homogeneous trees are studied. Essential selfadjointness of these matrices turns out to depend on the structure of the tree. If a tree has one end and infinitely many origin points the matrix is always essentially selfadjoint independently of the growth of its coefficients. In case a tree has one origin and infinitely many ends, the essential selfadjointness is equivalent to that of an ordinary Jacobi matrix obtained by restriction to the so called radial functions....

Joint subnormality of n-tuples and C₀-semigroups of composition operators on L²-spaces

Piotr Budzyński, Jan Stochel (2007)

Studia Mathematica

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)...

Joint subnormality of n-tuples and C₀-semigroups of composition operators on L²-spaces, II

Piotr Budzyński, Jan Stochel (2009)

Studia Mathematica

In the previous paper, we have characterized (joint) subnormality of a C₀-semigroup of composition operators on L²-space by positive definiteness of the Radon-Nikodym derivatives attached to it at each rational point. In the present paper, we show that in the case of C₀-groups of composition operators on L²-space the positive definiteness requirement can be replaced by a kind of consistency condition which seems to be simpler to work with. It turns out that the consistency condition also characterizes...

Jordan *-derivation pairs on standard operator algebras and related results

Dilian Yang (2005)

Colloquium Mathematicae

Motivated by Problem 2 in [2], Jordan *-derivation pairs and n-Jordan *-mappings are studied. From the results on these mappings, an affirmative answer to Problem 2 in [2] is given when E = F in (1) or when 𝓐 is unital. For the general case, we prove that every Jordan *-derivation pair is automatically real-linear. Furthermore, a characterization of a non-normal prime *-ring under some mild assumptions and a representation theorem for quasi-quadratic functionals are provided.

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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