Compactness Of Composition Operators Research Articles

Compact and Weakly Compact Composition Operators on BMOA

Any analytic map φ of the unit disc \({\mathbb{D}}\) into itself induces a composition operator C φ on BMOA, mapping \({f \mapsto f \circ \varphi}\), where BMOA is the Banach space of analytic functions \({f\colon \mathbb{D} \to \mathbb{C}}\) whose boundary values have bounded mean oscillation on the unit circle. We show that C φ is weakly compact on BMOA precisely when it is compact on BMOA, thus solving a question initially posed by Tjani and by Bourdon, Cima and Matheson in the special case of VMOA. As a crucial step of our argument we simplify the compactness criterion due to Smith for C φ on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. Additional equivalent compactness criteria are established. Furthermore, we prove the unexpected result that compactness of C φ on VMOA implies compactness even from the Bloch space into VMOA.

Compact composition operators on the Hardy--Orlicz and weighted Bergman-Orlicz spaces on the ball

Using recent characterizations of the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces on the ball, we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H^{\infty}. Then, we prove that, for each Koranyi region \Gamma, there exists a map \phi taking the ball into \Gamma such that, C_{\phi} is not compact on H^{\psi}\left(\mathbb{B}_{N}\right), when \psi grows fast. Finally, we give another characterization of the compactness of composition operator on weighted Bergman-Orlicz spaces in terms of an Orlicz Angular derivative-type condition. This extends (and simplify the proof of) a result by K. Zhu for the classical Bergman case. Moreover, we deduce that the compactness of composition operators on weighted Bergman-Orlicz spaces does not depend on the weight anymore, when the Orlicz function grows fast.

Composition operators between some weighted hardy spaces

Let D be the unit disc and H(D) be the set of all analytic functions on D. In [2], C. Cowen defined a space H = { f ∈ H ( D ) : F ( z ) = ∑ k = 0 ∞ a k ( z + 1 ) k , ∀ z ∈ D , ∥ f ∥ 2 = ∑ k = 0 ∞ | a k | 2 4 k < ∞ } In this article, the authors consider the similar Hardy spaces with arbitrary weights and discuss some properties of them. Boundedness and compactness of composition operators between such spaces are also studied.

Composition operators from Bloch type spaces into [formula omitted] type spaces

Bounded and compact composition operators Cφ, induced by an analytic self-map of the open unit disc D of the complex plane C, are characterized from the Bloch type spaces Bμ and Bμ,0 into the QK type spaces QK(p,q).

Composition operators on Bloch–Orlicz type spaces

In this paper we use Young’s functions to define the Bloch–Orlicz spaces as a generalization of Bloch spaces. We study continuity, boundedness from below and compactness of composition operators on Bloch–Orlicz spaces.

Composition operators and closures of some Möbius invariant spaces in the Bloch space

Boundedness and compactness of composition operators between the Bloch space and the closure of some Möbius invariant subspaces in the Bloch space are characterized.

Composition Operators on a Class of Analytic Function Spaces Related to Brennan’s Conjecture

Brennan’s conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk $${\mathbb{D}}$$ onto a simply connected domain G and −1/3 < p < 1, then 1/(τ′) p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces $${L^2_a(\mu _p)}$$ on G and prove that Brennan’s conjecture is equivalent to the existence of compact composition operators on these spaces for every simply connected domain G and all $${p\in(-1/3,1)}$$ . Motivated by this result, we study the boundedness and compactness of composition operators in this setting.

Lipschitz Continuous and Compact Composition Operators in Hyperbolic Classes

Natural metrics in the hyperbolic α-Bloch-, weighted Dirichlet- and Q p -classes are introduced, and these classes are shown to be complete metric spaces with respect to the corresponding metrics. Then Lipschitz continuous and compact composition operators C φ (f) = f ◦ φ acting from the hyperbolic α-Bloch-class to the hyperbolic weighted Dirichlet- or Q p -class are characterized by conditions depending on the symbol φ only.

Weighted composition operators on some function spaces of entire functions

We characterize the boundedness and compactness of the weighted composition operators acting between some Fock-type spaces. Motivated by some recent ideas by S. Steviic we also calculate the operator norm of some of these operators. Also we determine the form of the symbol which induces a bounded or compact composition operator between some Fock-type spaces.

COMPOSITION OPERATORS BELONGING TO SCHATTEN CLASS 𝒮p

AbstractWe investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener

We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.

Compact composition operators acting between weighted Bergman spaces of the unit ball

In this paper, we study a composition operator \({C_{\varphi}}\) on the weighted Bergman space \({A_{\alpha}^p(B)}\) of the unit ball B in \({{\mathbb{C}}^N}\) . Under a natural condition we estimate the essential norm of \({C_{\varphi}}\) . As a consequence of this estimate, we also give a function-theoretic characterization of \({\varphi}\) that induces a compact composition operator on \({A_{\alpha}^p(B)}\) .

Compact composition operators on BMOA(B n )

The paper gives some characterization theorems for the compact composition operator on some function spaces over the unit ball Bn in ℂn. Especially, it gives a characterization for compact composition operators on BMOA(Bn), which generalizes a result proved by Bourdon, Cima and Matheson for the case n = 1.

Compact composition operators on BMOA and the Bloch space

We give a new and simple compactness criterion for composition operators C ϕ on BMOA and the Bloch space in terms of the norms of ϕ n in the respective spaces.

Composition operators on some holomorphic Banach function spaces

In this paper, we study composition operators on some Möbius invariant Banach function spaces like Bloch and $F(p,q,s)$ spaces. We give a Carleson measure characterization on $F(p,q,s)$ spaces, then we use this Carleson measure characterization of the compact compositions on $F(p,q,s)$ spaces to show that every compact composition operator on $F(p,q,s)$ spaces is compact on a Bloch space. Also, we give conditions to clarify when the converse holds.

Composition operators between generally weighted Bloch spaces and $Q_{log}^q$ space

Let $\varphi$ be a holomorphic self-map of the open unit disk $D$ on the complex plane and $p, q>0.$ In this paper, the boundedness and compactness of composition operator $C_{\varphi}$ from generally weighted Bloch space $B^{p}_{\log}$ to $Q^{q}_{\log}$ are investigated.

Compact Composition Operators on Generalized Hardy Spaces

Abstract We investigate compact composition operators acting on generalized Hardy spaces 𝐻𝑤. In fact, we prove that if 𝑤 is a differentiable, subharmonic and strictly increasing function defined on [0, ∞), then 𝐶 φ is compact on the generalized Hardy spaces if and only if it is compact on the Hardy space 𝐻2.

Some examples of compact composition operators on [formula omitted

We construct, in an essentially explicit way, various composition operators on H 2 and study their compactness or their membership in the Schatten classes. We construct: non-compact composition operators on H 2 whose symbols have the same modulus on the boundary of D as symbols whose composition operators are in various Schatten classes S p with p > 2 ; compact composition operators which are in no Schatten class but whose symbols have the same modulus on the boundary of D as symbols whose associated composition operators are in S p for every p > 2 .

On the connected component of compact composition operators on the Hardy space

We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space H 2 . This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov–Clark measures.

Composition Operators on Orlicz–Lorentz Spaces

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