Composition Operators and Closures of Dirichlet Type Spaces Dμ in Bloch Type Spaces

We use a brief way to give various equivalent characterizations for the boundedness and the essential norm of the operator acting on Bloch-type spaces. At the same time, we use this method to easily get a known characterization for the operator on Bloch-type spaces. MSC:47B38, 26A24, 32H02, 47B33.

Let X be a finite or infinite dimensional complex Banach space. We characterize the bounded weighted composition operators between the Bloch type space and the Hardy space on the unit ball of X, extending several known results for finite dimensional domains.

For a wide class of radial weights we calculate the essential norm of a weighted composition operator \({uC_\varphi}\) on the weighted Banach spaces of analytic functions in terms of the analytic function \({u \colon \mathbb D \to \mathbb C}\) and the nth power of the analytic selfmap \({\varphi}\) of the open unit disc \({\mathbb D}\) . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general radial weights.

Composition Operators from Bloch Type Spaces to Hardy and Besov Spaces

While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces Bα with α ≠ 1 are the ones induced by a disc automorphism.

Let φ and ψ be analytic self-maps of the open unit disk D . Using pseudo-hyperbolic distance ρ(φ ,ψ) , we characterize the boundedness and compactness of the differences of generalized composition operators (C φ −Ch ψ ) f (z) = ∫ z 0 [ f ′(φ(ξ ))g(ξ )− f ′(ψ(ξ ))h(ξ )]dξ , z ∈ D between two Bloch-type spaces on D . The results generalize the corresponding results on the single generalized composition operator and on the differences of generalized composition operators on the Bloch space. Mathematics subject classification (2010): Primary 47B38; secondary 30H30.

Let $${\mathbf {B}}_{{\mathbb {X}}}$$ be the open unit ball of a complex Banach space $${\mathbb {X}}$$ , which may beinfinite dimensional. The weighted composition operator and weighted space defined on $${\mathbf {B}}_{{\mathbb {X}}}$$ are introduced. We obtain the boundedness and compactness of the weightedcomposition operator from the Bloch-type spaces to the weighted spaces, and some properties with the Bloch-type spaces are given. Our main results generalize theprevious works on the Euclidean unit ball $${\mathbb {B}}^n$$ to the case of $${\mathbf {B}}_{{\mathbb {X}}}$$ .

The boundedness and compactness of generalized weighted composition operators from Blochtype spaces and little Bloch-type spaces into weighted Zygmund spaces on the unit disc are characterized, in this paper.

When the weight $\mu$ is more general than normal, the complete characterizations in terms of the symbol $g$ and weights for the conditions of the boundedness and compactness of $T_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ and $S_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ are still unknown. Smith et al. firstly gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, continuing their lines of investigations, we give the complete characterizations of the conditions for the boundedness and compactness of Volterra type operators $T_g$ and $S_g$ between Bloch type spaces $\mathcal{B}^\infty_\nu$ and weighted Banach spaces $H^{\infty}_\nu$ with more general weights, which generalize their works.

We consider composition operators on Hardy spaces of a half-plane. We mainly study boundedness and compactness. We prove that on these spaces there are no compact composition operators.

In this paper we introduce natural metrics in the hyperbolic Bloch and Q K -type spaces with respect to which these spaces are complete. Moreover, Lipschitz continuous, bounded and compact superposition operators S ϕ from the hyperbolic Bloch type space to the hyperbolic Q K -type space are characterized by conditions depending only on the analytic symbol ϕ .

We give embedding theorems for Hardy–Orlicz spaces on the ball and then apply our results to the study of the boundedness and compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some Hardy–Orlicz spaces, different from H ∞, on which every composition operator is bounded.

We show that a restriction on the angular derivative of the inducing map does not determine compact composition operators on large weighted Hardy spaces, thus answering in the negative a question posed by T. Kriete.

Characterizations of Composition Operators on Bloch and Hardy Type Spaces

Consider an open unit disk D = z ∈ ℂ : z < 1 in the complex plane ℂ , ξ a holomorphic function on D , and ψ a holomorphic self-map of D . For an analytic function f , the weighted composition operator is denoted and defined as follows: W ξ , ψ f z = ξ z f ψ z . We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces.