Differences of generalized composition operators between Bloch type spaces

We characterize the closure of Dirichlet type spaces Dμ in Bloch type spaces. Moreover, we study the boundedness and compactness of composition operators from Bloch type spaces to closures of Dirichlet type spaces Dμ in Bloch type 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.

Characterizations of Composition Operators on Bloch and Hardy 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.

The s- Generalized Composition Operators from B(m,n) → Q Spaces gP

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.

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.

We give a new characterization for the boundedness of composi- tion operator followed by dierentiation operator acting on Bloch-type spaces and calculate its essential norm in terms of the n-th power of the induced an- alytic self-map on the unit disk. From which some sucient and necessary conditions of compactness of the operator follow immediately.

The aim of this paper is to show that univalent functions in several classical function spaces can be characterized by integral conditions involving the maximum modulus function. For a suitable choice of parameters, the established condition or its appropriate variant reduces to a known characterization of univalent functions in the Hardy or weighted Bergman space and gives a new characterization of univalent functions in several Mobius invariant function spaces, such as BMOA, Qp or the Bloch space. It is proved, for example, that univalent functions in the Dirichlet type space \( \mathcal{D}_{p + \alpha }^p \) are the same as the univalent functions in Hαp and Sαp if p ≥ 2. Moreover, it is shown that there is in a sense a much smaller Mobius invariant subspace of the Bloch space than Qp still containing all univalent Bloch functions.

We call a sequence {zn} in the open unit disk Δalmost thin if \({\mathop {\lim \sup }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 1,\) where Π(zn) is the product of the pseudo-hyperbolic distances between zn and the other elements in the sequence. We call the sequence thick if \({\mathop {\lim }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 0.\) In this paper, we use elementary complex variables methods to prove a sharper form of a characterization of the symbols of the isometric composition operators on the Bloch space proved by Martin and Vukotic in terms of the preimages of one-point sets. Specifically, we show that a bounded analytic function φ from Δ into itself has Bloch semi-norm equal to one (which is the largest possible) if and only if it is a conformal automorphism of Δ or if the preimage of each point of Δ contains a sequence along which the hyperbolic derivative of φ approaches one (condition (M’)). In particular, the preimage of each point of Δ is an almostthin sequence. Using similar techniques we also show that a bounded analytic function f is in the little Bloch space if and only if for each \(a\in \mathbb{C}\) such that \(f^{-1}(a)=\{z_{n}:n\in \mathbb{N}\}\) is infinite $$ {\mathop {\lim }\limits_{n \to \infty } }\left(1 - |z_{n} |^{2} \right)|f'{\left( {z_{n} } \right)}| = 0. $$ Consequently, an infinite Blaschke product whose left composition by a conformal automorphism of Δ is always a Blaschke product belongs to the little Bloch space if and only if the preimage of every point in Δ is a thick sequence. We give an example to show that this characterization does not extend to unbounded Bloch functions. We also prove an analogous characterization of the little normal functions. Finally, we give a sufficient condition on a holomorphic self-map of the polydisk Δn to be the symbol of an isometric composition operator on the Bloch space of Δn and discuss a higher-dimensional analogue of condition (M’).

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 $${\mathbf {B}}_{{\mathbb {X}}}$$ .

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.

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

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.