Parabolic composition operators on the ball

A class of linear fractional maps of the ball and their composition operators

Cyclic behavior of linear fractional composition operators in the unit ball of C-N

We give a classification, up to automorphisms, of hyperbolic linear fractional maps of the ball. We then show that this classification is very convenient to study the geometric properties of these maps, as well as the spectrum and the dynamics of the associated composition operators. We conclude by showing how these properties can be tranfered to composition operators associated to hyperbolic self-maps of the ball which are not linearfractional maps.

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Hyperbolic composition operators on the ball

Let $(T\_\lambda)\_{\lambda\in\Lambda}$ be a family of operators acting on a $F$-space $X$, where the parameter space $\Lambda$ is a subset of $\mathbb R^d$. We give sufficient conditions on the family to yield the existence of a vector $x\in X$ such that, for any $\lambda\in\Lambda$, the set $\big\{T\_\lambda^n x;\ n\geq 1\big\}$ is dense in $X$. We obtain results valid for any value of $d\geq 1$ whereas the previously known results where restricted to $d=1$. Our methods also shed new light on the one-dimensional case.

Common Hypercyclic Vectors for High-Dimensional Families of Operators

Inner Functions and Cyclic Composition Operators on H2(Bn)

In this paper, we study the Hankel operators belonging toE which denotes all of the essential Toeplitz operators, and show that the class of symbols of these Hankel operators is a Douglas algebra. Meanwhile, using the concept of essential commutant and some knowledge of the theory of function algebras, we solve a problem raised by Jose Barria and P. R. Halmos in [1].

Hankel operators in the set of essential Toeplitz operators

Let G be a discrete abelian group and (G, G+) an ordered group. Denote by (G, GF ) the minimal quasily ordered group containing (G, G+). In this paper, we show that the ideal of finite elements is exactly the kernel of the natural morphism between these two Toeplitz C∗-algebras. When G is countable, we show that if the direct sum of K-groups K0(T G+ )⊕K1(T G+ ) ∼= Z, then K0(T G+ ) ∼= Z. 1. Preliminaries and the main result Let G be a discrete abelian group and G+ a subset of G, (G, G+) is said to be a quasily ordered group, if 0 ∈ G+, G+ + G+ ⊆ G+, and G = G+ ∪ (−G+). If in addition, G+ = G+ ∩ (−G+) = {0}, then (G, G+) is referred to as an ordered group. In this section, we always fix an ordered group (G, G+). Let Ĝ be the dual group of G, Ĝ is compact and it is connected since G is torsion-free. When given the normal Haar measure, it is easy to show that { εx |x ∈ G } is an orthonormal basis for L(Ĝ), where εx is defined by εx(γ) = γ(x) for γ ∈ Ĝ. For any E ⊆ G, let H(E) be the closed subspace of L(Ĝ) generated by { εx |x ∈ E }; its projection is denoted by p . For any φ ∈ C(Ĝ), define T E φ on H(E) by T E φ (f) = p(φf) for f ∈ H(E). We say that T E φ is a Toeplitz operator ( relative to E ) with symbol φ. The C∗-algebra generated by {T E φ ∣∣ φ ∈ C(Ĝ) } is denoted by T E r , and is called the Toeplitz C∗-algebra with respect to E. By the StoneWeierstrass theorem, T E r is also generated by {T E εx ∣∣x ∈ G }. Now let (G, G+) be an ordered group, denote by F (G) = F (G+) ∪ (−F (G+)), where F (G+) = { x ∈ G+ ∣∣∀y ∈ G+ \ {0}, ∃n ∈ N, such that ny − x ∈ G+}. It is easy to show that F (G) is a subgroup of G and its elements are usually called the finite elements. Denote by K(F (G)), the closed two-sided ideal of T G+ r Received by the editors March 19, 1997 and, in revised form, June 3, 1997. 1991 Mathematics Subject Classification. Primary 47B35.

https://www.ams.org/proc/1999-127-02/S0002-9939-99-04774-7/S0002-9939-99-04774-7.pdf

Toeplitz c*-algebras on ordered groups and their ideals of finite elements

The main purpose of the present paper is to describe the Taylor joint spectra forn-tuples of double commuting hyponormal operators, and to study the representation of the joint spectra in terms of that ofn-tuples of commuting normal operators.

On the representation of the joint spectrum for a commutingn-tuple of non-normal operators

As we know, B.Sz-Nagy and C.Foins studied systematically contractions on Hilbert spaces and developed the harmonic analysis theory of operators on Hilbert spaces. Since 1950s, people paid great attention to the study of contractions on πk spaces. Only a few results have been obtained until today; in particular, the spectral theory of contractions on πk Spaces and corresponding harmonic analysis theory have left still unexplored. This paper, as a continuation of [1], [2], [6], in which the authors after discussing some problems such as the negative invariant subspaces and unitary dilations of contractions on complete spaces with indefinite metrics, establish the triangle model of contractions on πk spaces and furthermore, apply the triangle model to the study of spectral theory of contractions on πk spaces, which is essential to the harmonic analysis of operators on πk spaces.

Chen Xiaoman

Papers

Inner Functions and Cyclic Composition Operators on H2(Bn)

Hankel operators in the set of essential Toeplitz operators

Toeplitz c*-algebras on ordered groups and their ideals of finite elements

On the representation of the joint spectrum for a commutingn-tuple of non-normal operators

The spectral theory of contractions on Πk spaces