Topological Structure of the Space of Weighted Composition Operators on H ∞

We study properties of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology. Moreover, we characterize the compactness of differences of two weighted composition operators.     

Essential Norms of Composition Operators

We obtain simple estimates for the essential norm of a composition operator acting from the Hardy space H ^p to H ^q , p > q, in one or several variables. When p = and q = 2 our results give an exact formula for the essential norm.     

Differences of Composition Operators between Weighted Banach Spaces of Holomorphic Functions on the Unit Polydisk

We consider differences of composition operators between given weighted Banach spaces H ^∞ _v or H ⁰ _v of analytic functions defined on the unit polydisk D ^N with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. This paper is an extension of [6] where the one-dimensional case is treated. Received: May 15, 2007. Revised: October 8, 2007.     

Spectra of Composition Operators on BMOA

It is shown that if ϕ is a univalent self-map on the unit disk is not an automorphism and has a fixed point in and if the essential spectral radius of the composition operator C_ϕ on H² is different from zero, then the spectrum of C_ϕ on BMOA coincides with This answers in the affirmative a conjecture by MacCluer and Saxe.     

Composition Operators on Orlicz–Lorentz Spaces

In this paper, we study the boundedness and the compactness of composition operators on Orlicz–Lorentz spaces.      

Composition Operators on Small Spaces

We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro’s theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro’s theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.     

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Spectra of Compact Composition Operators Over Bounded Symmetric Domains

Let H²(D) denote the Hardy space of a bounded symmetric domain in its standard Harish-Chandra realization, and let be the weighted Bergman space with and where is a critical value depending on D. Suppose that is holomorphic. We show that if the composition operator defined by is compact (or, more generally, power-compact) on H²(D) or then has a unique fixed point z₀ in D. We then prove that the spectrum of as an operator on these function spaces is precisely the set consisting of 0, 1, and all possible products of eigenvalues of These results extend previous work by Caughran/Schwartz and MacCluer. As a corollary, we now have that MacCluers previous spectrum results on the unit ball B_n extend to H^p(n) (not only for p = 2 but for all p > 1) and (for p 1), where ⁿ is the polydisk in      

Composition Operators and Vector-valued BMOA

Analytic composition operators are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and C _φ is compact on BMOA, then C _φ is weakly compact on the X-valued space BMOA _C (X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.     

Holomorphic Operators Generating Dense Images

The existence of infinite dimensional closed linear spaces of holomorphic functions f on a domain G in the complex plane such that Tf has dense images on certain subsets of G, where T is a continuous linear operator, is analyzed. Necessary and sufficient conditions for T to have the latter property are provided and applied to obtain a number of concrete examples: infinite order differential operators, composition operators and multiplication operators, among others. This work was supported in part by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and by MEC DGES Grants MTM2006-13997-C02-01 and MTM2004-21420-E.     

Topological Components of the Set of Composition Operators on $$H^\infty (B_N)$$

This paper characterizes the component structure of the space of composition operators acting on , both in the operator norm topology and in the topology induced by the essential norm.     

Essential norm of differences of weighted composition operators between weighted-type spaces on the unit ball

We find an asymptotically equivalent expression to the essential norm of differences of weighted composition operators between weighted-type spaces of holomorphic functions on the unit ball in C^N. As a consequence we characterize the compactness of these operators. The boundedness of these operators is also characterized.     

Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions

We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact operators, on some uniform algebras of analytic functions. Even in the context of H^∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence of a general study of weighted composition operators.      

m-Isometric Commuting Tuples of Operators on a Hilbert Space

We consider a generalization of isometric Hilbert space operators to the multivariable setting. We study some of the basic properties of these tuples of commuting operators and we explore several examples. In particular, we show that the d-shift, which is important in the dilation theory of d-contractions (or row contractions), is a d-isometry. As an application of our techniques we prove a theorem about cyclic vectors in certain spaces of analytic functions that are properly contained in the Hardy space of the unit ball of .     

Linear-Fractional Composition Operators in Several Variables

We investigate properties of linear-fractional composition operators C_φ on Hardy and Bergman spaces of the ball in that are motivated by a formula for the self-commutator [C_φ^* ,C_φ]. In particular, we characterize when certain commutators [C_φ, C_σ] are compact, and give conditions under which is compact, where is multiplication by the monomial z^β. Our results allow us to determine when C_φ is essentially normal, for φ belonging to a large class of linear-fractional symbols.     

Composition operators and a pull-back measure formula

A pull-back measure formula obtained in some particular cases by E. A. Nordgren and this author is generalized in the framework of boundary measures for zero-free Nevanlinna class fuctions on the unit polydisk. The formula is used to characterize the zero-free Nevanlinna class functions which are solutions of Schröder's equation induced by a polydisk automorphism (i.e. to determine the zero-free functionsf belonging to the Nevanlinna class which are solutions of the functional equationf ° =f, for some constant ), thus generalizing earlier results obtained by R. Mortini and this author.     

How to get universal inner functions

We characterize a property that is useful for constructing large spaces of universal functions in a wide variety of settings. In particular, we show that given any sequence of automorphisms of the ball that admits a universal function , there exists a closed infinite dimensional subspace, generated by inner functions, that is isometric to . We put our results in the context of bounded symmetric domains.     

Isolation and Component Structure in Spaces of Composition Operators

We establish a condition that guarantees isolation in the space of composition operators acting between H^p(B_N) and H^q(B_N), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.     

On n-contractive and n-hypercontractive Operators

We consider in this paper the classes of n-hypercontractive Hilbert space operators, primarily weighted shifts, and obtain results for back step extensions of recursively generated subnormal weighted shifts and for perturbations in the first weight of the Bergman shift. We compare the results with those for the classes of k-hyponormal operators, and recapture, by an n-hypercontractive approach, a subnormality result originally proved in the k-hyponormal context.     

The Essential Norm of Hankel Operators on the Weighted Bergman Spaces with Exponential Type Weights

Let denote the closed subspace of consisting of analytic functions in the unit disc . For certain class of subharmonic functions and , it is shown that the essential norm of Hankel operator is comparable to the distance norm from H_f to compact Hankel operators.