Essential norm of composition operators between weighted Hardy spaces

An asymptotic formula for the essential norm of composition operators acting between two weighted Hardy spaces H_w1 and H_w2, where w₁ and w₂ are two admissible weight functions, is given. The boundedness of the operators is also characterized.     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball

We investigate the weighted composition operator from the weighted Bergman space into the weighted Hardy space on the unit ball. As a consequence of the investigation, we also give a characterization for the boundedness and compactness of the operator whose the target space is the Hardy space.     

Composition operators from large weighted Hardy spaces into the Dirichlet space

Given a large weighted Hardy space we show there exists a composition operatorC with that maps from that space into the unweighted Dirichlet space and lies in every Schatten p-class for 0<p<. This is in contrast to the situation in which the image space is a smaller weighted Dirichlet space. It is known that in that case it is not possible to find such a composition operator that is bounded.This research was supported in part by a summer stipend from Bellarmine College.     

Essential norm of the difference of composition operators on Bloch space

Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C _ϕ , C _ψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators C _ϕ −C _ψ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.     

Essential norm of composition operators on the Hardy space H 1 and the weighted Bergman spaces {A_{\alpha}^{p}} on the ball

We estimate the essential norm of a composition operator acting on the Hardy space H ¹ and the weighted Bergman spaces ${A_{\alpha}^{p}}$ on the unit ball. In passing, we recover (and somehow simplify the proof of) parts of the recent article by Demazeux, dealing with the same question for H ¹ of the unit disc. We also estimate the essential norm of a composition operator acting on ${A_{\alpha}^{p}}$ in terms of the angular derivatives of ${\phi}$ , under a mild condition on ${\phi}$ .     

Superposition operators between the Bloch space and Bergman spaces

Using a geometric method, we characterize all entire functions that transform the Bloch space into a Bergman space by superposition in terms of their order and type. We also prove that all superposition operators induced by such entire functions act boundedly. Similar results hold for superpositions from BMOA into Bergman spaces and from the Bloch space into certain weighted Hardy spaces.     

Essential norms of weighted composition operators from weighted Bergman space to mixed-norm space on the unit ball

In this paper, we characterize the boundedness and compactness of the weighted composition operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.     

Composition operators with maximal norm on weighted Bergman spaces

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces (in particular, on the space ) is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space , where every inner function induces a composition operator with maximal norm.

     

Compact differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces

We characterize the compactness of differences of weighted composition operators from the weighted Bergman space , 0 < p < ∞, α > −1, to the weighted-type space of analytic functions on the open unit disk D in terms of inducing symbols and . For the case 1 < p < ∞ we find an asymptotically equivalent expression to the essential norm of these operators.     

Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces

We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.     

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.     

Weighted composition operators from Bergman-type spaces into Bloch spaces

Let ϕ be an analytic self-map and u be a fixed analytic function on the open unit disk D in the complex plane ℂ. The weighted composition operator is defined by

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Compact weighted composition operators on the Hardy space

A weighted composition operator takes an analytic map on the open unit disk of the complex plane to the analytic map , where is an analytic map of the open unit disk into itself and is an analytic map on the open unit disk. This paper studies how the compactness of depends on the interaction between the two maps and .