Adjoints of composition operators with rational symbol

Building on techniques developed by Cowen and Gallardo-Gutiérrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space H². We consider some specific examples, comparing our formula with several results that were previously known.     

Adjoints of composition operators on Hardy spaces of the half-plane

Building on techniques used in the case of the disc, we use a variety of methods to develop formulae for the adjoints of composition operators on Hardy spaces of the upper half-plane. In doing so, we prove a slight extension of a known necessary condition for the boundedness of such operators, and use it to provide a complete classification of the bounded composition operators with rational symbol. We then consider some specific examples, comparing our formulae with each other, and with other easily deduced formulae for simple cases.     

Adjoints of a class of composition operators

Adjoints of certain operators of composition type are calculated. Specifically, on the classical Hardy space of the open unit disk operators of the form are considered, where is a finite Blaschke product. is obtained as a finite linear combination of operators of the form where and are rational functions, are associated Toeplitz operators and is defined by

     

Adjoints of composition operators on Hilbert spaces of analytic functions

We observe that a formula for the adjoint of a composition operator, known only for special symbols in some spaces of analytic functions, actually holds for every admissible symbol and in any Hilbert space of analytic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form.     

A new class of operators and a description of adjoints of composition operators

Starting with a general formula, precise but difficult to use, for the adjoint of a composition operator on a functional Hilbert space, we compute an explicit formula on the classical Hardy Hilbert space for the adjoint of a composition operator with rational symbol. To provide a foundation for this formula, we study an extension to the definitions of composition, weighted composition, and Toeplitz operators to include symbols that are multiple-valued functions. These definitions can be made on any Banach space of analytic functions on a plane domain, but in this work, our attention is focused on the basic properties needed for the application to operators on the standard Hardy and Bergman Hilbert spaces on the unit disk.     

Adjoints of rationally induced weighted composition operators on the Hardy,Bergman and Dirichlet spaces

We determine the adjoint of a multiplication operator with rational symbol u acting on various spaces of analytic functions, in which the denominator of u is a product of distinct linear factors. We use the results to represent the adjoints of weighted composition operators with rational symbols on the Hardy, Bergman and Dirichlet spaces.     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

Products of integral-type operators and composition operators between Bloch-type spaces

A complete picture on the boundedness and compactness of the products of integral-type operators and composition operators between Bloch-type spaces of holomorphic functions on the unit disk is given in this paper.     

Separating classes of composition operators via subnormal condition

Several classes have been considered to study the weak subnormalities of Hilbert space operators. One of them is -hypnormality, which comes from the Bram-Halmos criterion for subnormal operators. In this note we consider -hyponormality, which is the parallel version corresponding to the Embry characterization for subnormal operators. We characterize -hyponormality of composition operators via -th Radon-Nikodym derivatives and present some examples to distinguish the classes.

     

Compact composition operators on the Smirnov class

We show that a composition operator on the Smirnov class is compact if and only if it is compact on some (equivalently: every) Hardy space for . Along the way we show that for composition operators on both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.

     

Norm-attaining composition operators on the Bloch spaces

We prove that every composition operator C? on the Bloch space (modulo constant functions) attains its norm and characterize the norm-attaining composition operators on the little Bloch space (modulo constant functions). We also identify the extremal functions for ‖C?‖ in both cases.     

Leibenzon's backward shift and composition operators

We apply Leibenzon's backward shift to show that the composition operator on the unit ball of always maps the weighted Hardy space into the Hardy class .

     

Compact differences of composition operators

A characterization of compact difference is given for composition operators acting on the standard weighted Bergman spaces and necessary conditions are given on a larger scale of weighted Dirichlet spaces. Conditions are given under which a composition operator can be written as a finite sum of composition operators modulo the compacts. The additive structure of the space of composition operators modulo the compact operators is investigated further and a sufficient condition is given to insure that two composition operators lie in the same component.     

The identity is isolated among composition operators

Let be the Banach algebra of bounded holomorphic functions on the open unit ball of a Banach space. We show that the identity operator is an isolated point in the space of composition operators on . This answers a conjecture of Aron, Galindo and Lindström.

     

Derivative-free characterizations of bounded composition operators between Lipschitz spaces

We prove that maps into if and only if belongs to . In the case β < 1, we give another two equivalent conditions. Supported by MNZŽS Serbia, Project No. ON144010.     

Norms of composition operators with rational symbol

We compute the norms of composition operators with rational symbols that satisfy certain properties, extending Christopher Hammond's methods on operators with linear fractional symbols. This leads to a host of new examples of composition operators whose norms are calculable.     

Characterization of composition operators with closed range for one-dimensional smooth symbols

We give a full characterization of smooth symbols ψ:R→R$\psi :\mathbb{R}\to \mathbb{R}$ for which the composition operator _Cψ:C^∞(R)→C^∞(R)$C_{\psi }:C^{\infty }(\mathbb{R})\to C^{\infty }(\mathbb{R})$, F?F°ψ$F?F°\psi$ has closed range. This generalizes in a special case the result of Kenessey and Wengenroth who gave such a characterization for smooth injective   symbols ψ:R→R^d$\psi :\mathbb{R}\to {\mathbb{R}}^d$.     

Extreme points of the closed convex hull of composition operators

We study the extreme points of the closed convex hull of the set of all composition operators on the space of bounded analytic functions and the disk algebra.