Linear Fractional Composition Operators Research Articles

Cyclic behavior of linear fractional composition operators in the unit ball of [formula omitted

We characterize the cyclicity and hypercyclicity of composition operators induced by linear fractional self-maps of B N on the Hardy space H 2 ( B N ) based on the classification of linear fractional maps given by Bisi and Bracci.

The Fredholm Index for Elements of Toeplitz-Composition C*-Algebras

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse.

Linear-Fractional Composition Operators in Several Variables

We investigate properties of linear-fractional composition operators Cφ on Hardy and Bergman spaces of the ball in \(\mathbb{C}^N \) that are motivated by a formula for the self-commutator [Cφ* ,Cφ]. In particular, we characterize when certain commutators [Cφ, Cσ] are compact, and give conditions under which \(\left[ {T_{z^\beta } ^* ,C_\varphi } \right]\) is compact, where \(T_{z^\beta }\) is multiplication by the monomial zβ. Our results allow us to determine when Cφ is essentially normal, for φ belonging to a large class of linear-fractional symbols.

The role of the spectrum in the cyclic behavior of composition operators

A bounded operator T acting on a Hilbert space H is called cyclic if there is a vector x such that the linear span of the orbit {Tnx:n≥0} is dense in H. If the scalar multiples of the orbit are dense, then T is called supercyclic. Finally, if the orbit itself is dense, then T is called hypercyclic. We completely characterize the cyclicity, the supercyclicity and the hypercyclicity of scalar multiples of composition operators, whose symbols are linear fractional maps, acting on weighted Dirichlet spaces. Particular instances of these spaces are the Bergman space, the Hardy space, and the Dirichlet space. Thus, we complete earlier work on cyclicity of linear fractional composition operators on these spaces. In this way, we find exactly the spaces in which these composition operators fail to be cyclic, supercyclic or hypercyclic. Consequently, we answer some open questions posed by Zorboska. In almost all the cases, the cut-off of cyclicity, supercyclicity or hypercyclicity of scalar multiples is determined by the spectrum. We will find that the Dirichlet space plays a critical role in the cut-off.

Norms of linear-fractional composition operators

We obtain a representation for the norm of the composition operator C Φ on the Hardy space H 2 whenever Φ is a linear-fractional mapping of the form Φ(z) = b/(cz + d). The representation shows that, for such mappings Φ, the norm of C Φ always exceeds the essential norm of C Φ . Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form Φ(z) = sz + t has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers s and t, Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of C 1/(2-z) is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator C Φ , for which ∥C Φ ∥ > ∥C Φ ∥ e , an equation whose maximum (real) solution is ∥C Φ ∥ 2 . Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.

Adjoints of linear fractional composition operators on the Dirichlet space

The adjoint of a linear fractional composition operator acting on the classical Dirichlet space is expressed as another linear fractional composition operator plus a two rank operator. The key point is that, in the Dirichlet space modulo constant functions, many linear fractional composition operators are similar to multiplication operators and, thus, normal. As a particular application, we can easily deduce the spectrum of each linear fractional composition operator acting on such spaces. Even the norm of each linear fractional composition operator is computed on the Dirichlet space modulo constant functions. It is also shown that all this work can be carried out in the Hardy space of the upper half plane.

Which linear-fractional composition operators are essentially normal?

We characterize the essentially normal composition operators induced on the Hardy space H 2 by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H 2 that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.

Components of linear-fractional composition operators

We determine when two linear-fractional composition operators on the Hardy space H 2 belong to the same component in the collection of all composition operators on H 2. We show that two such composition operators in the same component may fail to have compact difference, which answers a question raised by Joel Shapiro and Carl Sundberg.

Relating composition operators on different weighted Hardy spaces

In a 1988 paper, Cowen found a formula expressing the adjoint of any linear fractional composition operator on the Hardy space as a product of Toeplitz operators and another linear fractional composition operator. In this paper, we use Cowen's adjoint formula to give a unitary equivalence relating composition operators on different weighted Hardy spaces. This result is then applied to some composition operators on the Sa spaces. We find the spectrum of any linear fractional composition operator whose symbol has exactly one fixed point of multiplicity one on the unit circle.

Cyclic phenomena for composition operators

Introduction Preliminaries Linear-fractional composition operators Linear-fractional models The hyperbolic and parabolic models Cyclicity: Parabolic nonautomorphism case Endnotes References.

Linear fractional composition operators on H2

If ϕ is an analytic function mapping the unit diskD into itself, the composition operatorC ϕ is the operator onH 2 given byC ϕf=foϕ. The structure of the composition operatorC ϕ is usually complex, even if the function ϕ is fairly simple. In this paper, we consider composition operators whose symbol ϕ is a linear fractional transformation mapping the disk into itself. That is, we will assume throughout that $$\varphi \left( z \right) = \frac{{az + b}}{{cz + d}}$$ for some complex numbersa, b, c, d such that ϕ maps the unit diskD into itself. For this restricted class of examples, we address some of the basic questions of interest to operator theorists, including the computation of the adjoint.