Bloch Space Research Articles

The iterates of composition operators on Banach spaces of holomorphic functions

In this paper we study uniform convergence, strong convergence, weak convergence, and ergodicity of the iterates of composition operators Cφ on various Banach spaces of holomorphic functions on the unit disk, such as Bergman spaces, Dirichlet spaces, weighted Banach spaces with sup-norm, and weighted Bloch spaces. For many spaces, the following results are proved:(i)the iterates Cφn do not converge in the weak operator topology and Cφ is mean ergodic if φ is an elliptic automorphism,(ii)Cφn converge uniformly if the Denjoy-Wolff point of φ is in D,(iii)Cφ is not mean ergodic if the Denjoy-Wolff point of φ lies on the boundary ∂D.

Ergodic properties of composition operators on Banach spaces of analytic functions

The composition operators defined on little Bloch spaces, Bergman spaces, Hardy spaces or little weighted Bergman spaces of infinite type, when well defined, are shown to be mean ergodic if and only if they are power bounded if and only if the symbol has an interior fixed point. For these operators uniform mean ergodicity is equivalent to quasicompactness in the sense of Yosida and Kakutani.

Weighted composition operators from the Bloch space tonth weighted-typespaces

In this work, we characterize the boundedness of weighted composition operators from the Bloch space and the little Bloch space to $n$th weighted-type spaces. Some estimates for the essential norm of these operators are also given. As a corollary, we obtain some characterizations for the compactness of weighted composition operators from the Bloch space and the little Bloch space to $n$th weighted-type spaces.

Weighted Composition Operators from Banach Spaces of Holomorphic Functions to Weighted-Type Banach Spaces on the Unit Ball in $$\mathbb {C}^n$$

Let X be a Banach space of holomorphic functions on the unit ball $$\mathbb {B}_n$$ in $$\mathbb {C}^n$$ whose point-evaluation functionals are bounded. In this work, we characterize the bounded weighted composition operators from X into a weighted-type Banach space $$H^\infty _\mu (\mathbb {B}_n)$$, where the weight $$\mu $$ is an arbitrary positive continuous function on $$\mathbb {B}_n$$. We determine the norm of such operators in terms of the norm of the point-evaluation functionals. Under some restrictions on X, we characterize the compact weighted composition operators mapping X into $$H^\infty _\mu (\mathbb {B}_n)$$. Under an alternative set of conditions, we provide essential norm estimates. We apply our results to the cases when X is the Hardy space $$H^p(\mathbb {B}_n)$$, the weighted Bergman space $$A_\alpha ^p(\mathbb {B}_n)$$ for $$\alpha >-1$$ and $$1\le p<\infty $$, the Bloch space $$\mathcal {B}$$ and the little Bloch space $$\mathcal {B}_0$$. In all these cases we obtain precise formulas of the essential norm.

Estimates of Norm and Essential norm of Differences of Differentiation Composition Operators on Weighted Bloch Spaces

Norm and essential norm of differences of differentiation composition operators between Bloch spaces have been estimated in this paper. As a result, we find characterizations for boundedness and compactness of these operators.

Atomic decomposition and weak factorization for Bergman–Orlicz spaces

For $\mathbb B^n$ the unit ball of $\mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L^\Phi_\alpha(\mathbb B^n)$, which are generalizations of classical Bergman spaces. We obtain atomic decomposition for functions in the Bergman-Orlicz space $\mathcal A^\Phi_\alpha (\mathbb B^n)$ where $\Phi$ is either convex or concave growth function. We then prove weak factorization theorems involving the Bloch space and a Bergman-Orlicz space and also weak factorization theorems involving two Bergman-Orlicz spaces.

Hypercyclicity of weighted composition operators on the weighted little Bloch space

We characterize the hypercyclicity of weighted composition operators on the weighted little Bloch space B0α, α>0.

Weighted composition operators between weighted-type spaces and the Bloch space and BMOA

Let $${\mathbb {D}}$$ denote the open unit disk in $${\mathbb {C}}$$. For an integer $$n\ge 0$$, let $$V_n$$ be the space defined recursively by $$\begin{aligned} V_0=\big \{f:{\mathbb {D}}\rightarrow {\mathbb {C}}: f \text { analytic}, |f(z)|=O\big ((1-|z|)^{-1}\big )\big \}, \end{aligned}$$and for $$n\ge 1$$, $$f\in V_n$$ if and only if $$f'\in V_{n-1}$$. In this work, we characterize the bounded and the compact weighted composition operators between $$V_n$$ and the Bloch space and the space BMOA of analytic functions of bounded mean oscillation, respectively. We also show that the bounded (respectively, compact) weighted composition operators mapping the space BMOA into $$V_n$$ are precisely the same as the bounded (respectively, compact) weighted composition operators mapping the Bloch space into $$V_n$$.

POWERS OF COMPOSITION OPERATORS: ASYMPTOTIC BEHAVIOUR ON BERGMAN, DIRICHLET AND BLOCH SPACES

We study the asymptotic behaviour of the powers of a composition operator on various Banach spaces of holomorphic functions on the disc, namely, standard weighted Bergman spaces (finite and infinite order), Bloch space, little Bloch space, Bloch-type space and Dirichlet space. Moreover, we give a complete characterization of those composition operators that are similar to an isometry on these various Banach spaces. We conclude by studying the asymptotic behaviour of semigroups of composition operators on these various Banach spaces.

Some applications for generalized fractional operators in analytic functions spaces

In this study a new generalization for operators of two parameters type of fractional in the unit disk is proposed. The fractional operators in this generalization are in the Srivastava-Owa sense. Concerning with the related applications, the generalized Gauss hypergeometric function is introduced. Further, some boundedness properties on Bloch space are also discussed.

Harmonic Besov spaces with small exponents

ABSTRACTWe study harmonic Besov spaces on the unit ball of , where 0<p<1 and . We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We show that the dual of harmonic Besov space is weighted Bloch space under certain volume integral pairing for 0<p<1 and . Our other results are about growth at the boundary and atomic decomposition.

Metric Spaces of Bounded Analytical Functions

In this paper, we consider classes of analytical functions that map the unit disk into itself. Functions of these classes can be described in terms of hyperbolic derivative and hyperbolic metric. Under an appropriate choice of the corresponding metrics, these classes are metric spaces. Functions of the hyperbolic classes considered generate composition operators from the Bloch space into classical spaces of analytical functions in the unit disk.

The Products of Differentiation and Composition Operators from Logarithmic Bloch Spaces to $$\mu $$-Bloch Spaces

In this paper, we present the new characterizations, in terms of the sequence $$\{z^j\}_{j=1}^\infty $$, about the boundedness and essential norm estimation for the products of differentiation and composition operators from logarithmic Bloch spaces to $$\mu $$-Bloch spaces on the unit disk. Then our main results are applied to show the boundedness of a non-trivial product operator $$C_\varphi D^m$$ with $$\varphi (z)=z^2$$.

Weighted fractional composition operators on certain function spaces

In this paper, we give some characterizations for the boundedness of weighted fractional composition operator [Formula: see text] from [Formula: see text]-Bloch spaces into weighted type spaces by deriving the bounds of its norm. Also, estimates for essential norm are obtained which gives necessary and sufficient conditions for the compactness of the operator [Formula: see text].

Composition Operators and the Closure of Morrey Space in the Bloch Space

In this paper, we characterize the closure of the Morrey space in the Bloch space. Furthermore, the boundedness and compactness of composition operators from the Bloch space to the closure of the Morrey space in the Bloch space are investigated.

Cauchy and Bergman projection, sharp gradient estimates and certain operator norm equalities

ABSTRACTWe get sharp pointwise estimates for the gradient of Pf, where P is Bergman projection in terms of -norm of function f defined in . Using limiting argument we transfer this result to Cauchy projection on and hence, the optimal gradient estimates of solution of -problem, thus extending results from Kalaj, Vujadinović [Norm of the Bergman projection onto the Bloch space. J Oper Theory. 2015;73(1):113–126], Kalaj, Marković [Optimal estimates for the gradient of harmonic functions in the unit disk. Complex Anal Oper Theory. 2013;7:1167–1183], Melentijević [Norm of the Bergman projection onto the Bloch space with -invariant gradient norm. arXiv 1711.08719[math.CV]]. As corollaries we get the sharp gradient estimate of a function in Hardy and Bergman spaces and exact norms of Cauchy projection acting into the Bloch space equipped with several (quasi)-norms.

Some Properties of Linear Operators Defined by Hypergeometric Functions

This paper is concerned with the connection of certain operators Φb,c, defined by hypergeometric functions, with the α-logarithmic Bloch spaces Blog,α of analytic functions.

Weighted Composition Operators on Iterated Weighted-Type Banach Spaces of Analytic Functions

We study a class $$\{V_n : n\ge 0\}$$ of iterated weighted-type Banach spaces of analytic functions on the open unit disk of which the Bloch space and the Zygmund space are special cases. We characterize the bounded and the compact weighted composition operators on $$V_n$$ thereby extending several known results in the literature. Lastly, we characterize the invertible weighted composition operators on $$V_n$$ and prove that a composition operator on $$V_n$$ is an isometry if and only if its symbol is a rotation.

Toeplitz Operator and Carleson Measure on Weighted Bloch Spaces

In this paper, we consider Toeplitz operator acting on weighted Bloch spaces. Meanwhile, the inclusion map from weighted Bloch spaces into tent space is also investigated.

Norm of the Bergman projection onto the Bloch space with M-invariant gradient norm

The value of the operator norm of Bergman projections from $L^{\infty}(\mathbb{B}^n)$ to Bloch space is found in \cite{KalajMarkovic2014}. The authors of mentioned paper proposed the problem of calculating the norm of the same class of operators with different norm on the Bloch space - $\mathcal{M}$-invariant gradient. Here we prove that it has the conjectured value.