Compact Differences Of Composition Operators Research Articles

Compact difference of composition operators on the Hardy spaces

Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of certain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.

Compact difference of composition operators with smooth symbols on the unit ball

A long-standing problem raised by Shapiro and Sundberg in 1990, the characterization for compact differences of composition operators acting on the Hardy space over the unit disk, has been recently obtained in terms of certain Carleson measures ([4]). The function theoretic characterization for compact differences still remains open in the Hardy space case even on the unit disc and the situation is much more complicated for the several variables case. In this article, we investigate a function theoretic characterization of the compact difference on the Hardy or the Bergman spaces over the unit ball. The condition ρ(φ(z),ψ(z))→0 as max⁡{|φ(z)|,|ψ(z)|}→1 is shown to be sufficient for the difference, Cφ−Cψ, of two composition operators to be compact on the Hardy or the Bergman spaces over the unit ball when each single composition operator is bounded. We show that this condition is also a necessary condition if the symbols are of class Lip1(B).

Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

Bounded and compact differences of two composition operators acting from the weighted Bergman space A^p_omega to the Lebesgue space L^q_nu , where 0<q<p<infty and omega belongs to the class of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for A^p_omega , with p>q and , involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space A^p_alpha with -1<alpha <infty to the setting of doubling weights. The case is also briefly discussed and an open problem concerning this case is posed.

Difference of composition operators over the half-plane

To overcome the unboundedness of the half-plane, we use Khinchine’s inequality and atom decomposition techniques to provide joint Carleson measure characterizations when the difference of composition operators is bounded or compact from standard weighted Bergman spaces to Lebesgue spaces over the half-plane for all index choices. For applications, we obtain direct analytic characterizations of the bounded and compact differences of composition operators on such spaces. This paper concludes with a joint Carleson measure characterization when the difference of composition operators is Hilbert-Schmidt.

Difference of weighted composition operators

We obtain complete characterizations in terms of Carleson measures for bounded/compact differences of weighted composition operators acting on the standard weighted Bergman spaces over the unit disk. Unlike the known results, we allow the weight functions to be non-holomorphic and unbounded. As a consequence we obtain a compactness characterization for differences of unweighted composition operators acting on the Hardy spaces in terms of Carleson measures and, as a nontrivial application of this, we show that compact differences of composition operators with univalent symbols on the Hardy spaces are exactly the same as those on the weighted Bergman spaces. As another application, we show that an earlier characterization due to Acharyya and Wu for compact differences of weighted composition operators with bounded holomorphic weights does not extend to the case of non-holomorphic weights. We also include some explicit examples related to our results.

Compact differences of composition operators on large weighted Bergman spaces

While there have been extensive studies regarding the theory of composition operators in standard Bergman spaces, there have not been many results pertaining to large Bergman spaces due to a lack of useful tools. In this paper, we obtain a complete characterization of the compact differences of composition operators in Bergman spaces with weight of ω(z)=e−11−|z| using a newly defined Riemannian distance. Moreover, we discuss the compact differences of composition operators in Bergman spaces with general weights.

Compact differences of composition operators induced by smooth self-maps of the complex unit ball

Let φ and ψ be holomorphic self-maps of the unit ball B N , which are sufficiently smooth on the sphere ∂ B N . In this paper, the compact differences of composition operators induced by φ and ψ on the weighted Bergman spaces or Hardy spaces are studied. The main result concerning φ and ψ with the same boundary data is a nontrivial generalization of the corresponding results in the case of one complex variable. It reveals the deep connection between the induced maps φ and ψ on the sphere.

Compact differences of composition operators on weighted Dirichlet spaces

AbstractHere we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S 2, the space of analytic functions whose first derivative is in H 2, and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.

Compact Differences of Composition Operators on the Bergman Spaces Over the Ball

The compact differences of composition operators acting on the weighted L 2-Bergman space over the unit disk is characterized by the angular derivative cancellation property and due to Moorhouse. In this paper we extend Moorhouse’s characterization, as well as some related results, to the ball and, at the same time, to the weighted L p -Bergman space for the full range of p.

Compact Differences of Composition Operators over Polydisks

Moorhouse characterized compact differences of composition operators acting on a weighted Bergman space over the unit disk of the complex plane. She also found a sufficient condition for a single composition operator to be a compact perturbation of the sum of given finitely many composition operators and studied the role of second order data in determining compact differences. In this paper, based on the characterizations due to Stessin and Zhu, of boundedness and compactness of composition operators acting from a weighted Bergman space into another, we obtain the polydisk analogues of Moorhouse’s results through a different approach in main steps. In addition we find a necessary coefficient relation for compact combinations which was first noticed on the disk by Kriete and Moorhouse.

Compact differences of composition operators on Holomorphic Function Spaces in the unit ball

We find a lower bound for the essential norm of the difference of two composition operators acting on H 2(B N) or A 2 s (B N) ( s > −1). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.

Compact Differences of Composition Operators in Several Variables

When φ and ψ are linear–fractional self-maps of the unit ball BN in \({{\mathbb C}^N,N\geq 1}\), we show that the difference \({C_{\varphi}-C_{\psi}}\) cannot be non-trivially compact on either the Hardy space H2(BN) or any weighted Bergman space \({A^2_{\alpha}(B_N)}\). Our arguments emphasize geometrical properties of the inducing maps φ and ψ.

Compact Differences of Composition Operators on Bloch and Lipschitz Spaces

We consider the difference T = Cφ − Cψ of two analytic composition operators in the unit disc. We characterize the compactness and weak compactness of T on the standard Bloch space, improving an earlier result by Hosokawa and Ohno. We also characterize the compactness and weak compactness of T on analytic Lipschitz spaces. These characterizations are derived from a general result dealing with differences of weighted composition operators on weighted Banach spaces of analytic functions. We also make complementary remarks on the compactness properties of a single composition operator on the Lipschitz spaces and answer a question of Cowen and MacCluer on the boundedness of such an operator.

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.