Analytic Self-map Research Articles

A new class of Carleson measures and integral operators on Bergman spaces

Let n be a positive integer and u=(u0,u1,…,un) with uk∈H(D) for 0≤k≤n, where H(D) is the space of analytic functions in the unit disk D. For 0<p,q<∞, we introduced a new class of (u,p,q)-Sobolev Carleson measure for the Bergman space Ap, i.e. a positive Borel measure μ on D, for which there exists a constant C>0 such that∫D|u0(z)f(z)+u1(z)f′(z)+⋯+un(z)f(n)(z)|qdμ(z)≤C‖f‖pq for all f∈Ap. Using Sobolev Carleson measures, we characterized the boundedness and compactness of the generalized Volterra-type operator Ig(n) acting on Bergman space to another, which is represented asIg(n)f=In(fg0+f′g1+⋯+f(n−1)gn−1), here g=(g0,⋯,gn−1) with gk∈H(D) for 0≤k≤n−1 and (If)(z)=∫0zf(w)dw is the usual integration operator. This operator is a generalization of the operator introduced by Chalmoukis in [5]. As a consequence, we obtain conditions for certain linear differential equations to have solutions in Bergman spaces. Moreover, we study the boundedness, compactness and Hilbert-Schmidtness of the following sums of generalized weighted composition operators: Lu,φ(n)=∑k=0nWuk,φ(k), Where φ is an analytic self-map of D and Wuk,φf=uk⋅f(k)∘φ.

A Linear Composition Operator on the Bloch Space

Let n∈N0, ψ be an analytic self-map on D and u be an analytic function on D. The single operator Du,ψn acting on various spaces of analytic functions has been a subject of investigation for many years. It is defined as (Du,ψnf)(z)=u(z)f(n)(ψ(z)),f∈H(D). However, the study of the operator Pu→,ψk, which represents a finite sum of these operators with varying orders, remains incomplete. The boundedness, compactness and essential norm of the operator Pu→,ψk on the Bloch space are investigated in this paper, and several characterizations for these properties are provided.

Volterra-Composition Operators Acting on $S^{p}$ Spaces and Weighted Zygmund Spaces

Let $\varphi$ be an analytic selfmap of the open unit disk $\mathbb{D}$ and $g$ be an analytic function on $\mathbb{D}$. The Volterra-type composition operators induced by the maps $g$ and $\varphi$ are defined as $$\left(I_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f^{\prime}(\varphi(\zeta)) g(\zeta) d\zeta \hspace{0.07in} \text{and} \hspace{0.07in} \left(T_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f(\varphi(\zeta)) g^{\prime}(\zeta) d\zeta .$$For $1\leq p&lt;\infty$, $S^p(\mathbb{D})$ is the space of all analytic functions on $\mathbb{D}$ whose first derivative $f^{\prime}$ lies in the Hardy space $H^p(\mathbb{D})$, endowed with the norm $\displaystyle \|f\|_{S^p}=|f(0)|+\|f^{\prime}\|_{H^p}$. Let $\mu: (0,1] \rightarrow (0, \infty)$ be a positive continuous function on $\mathbb{D}$ such that for $z\in\mathbb{D}$ we define $\mu(z) = \mu(|z|)$. The weighted Zygmund space $\mathcal{Z}_{\mu}(\mathbb{D})$ is the space of all analytic functions $f$ on $\mathbb{D}$ such that $\sup_{z\in \mathbb{D}} \mu(z) |f^{\prime\prime}(z)|$ is finite. In this paper, we characterize the boundedness and compactness of the Volterra-type composition operators that act between $S^p$ spaces and weighted Zygmund spaces.

Complete Continuity of Composition-Differentiation Operators on the Hardy Space H 1

We study composition-differentiation operators on the Hardy space H 1 on the unit disk. We prove that if φ is an analytic self-map of the unit disk such that the composition-differentiation operator induced by φ is bounded on the Hardy space H 1 , then it is completely continuous. This result is stronger than the similar result for composition operators which says that the composition operator induced by φ is completely continuous if and only if φ e i θ &lt; 1 almost everywhere on the unit circle.

Composition-differentiation operators acting on certain Hilbert spaces of analytic functions

We study composition-differentiation operators acting on the Bergman and Dirichlet space of the open unit disk. We first characterize the compactness of composition-differentiation operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\varphi$ on the open unit disk $\mathbb{D}$, the induced composition-differentiation operator is bounded with dense range if and only if $\varphi$ is univalent and the polynomials are dense in the Bergman space on $\Omega:=\varphi(\mathbb{D})$.

Refined Bohr inequalities for certain classes of functions: analytic, univalent, and convex

AbstractIn this article, we prove several refined versions of the classical Bohr inequality for the class of analytic self-mappings on the unit disk $ \mathbb {D} $ , class of analytic functions $ f $ defined on $ \mathbb {D} $ such that $\mathrm {Re}\left (f(z)\right )&lt;1 $ , and class of subordination to a function g in $ \mathbb {D} $ . Consequently, the main results of this article are established as certainly improved versions of several existing results. All the results are proved to be sharp.

Generalized Volterra type integral operators on large Bergman spaces

Let ϕ be an analytic self-map of the open unit disk D and g analytic in D. We characterize boundedness and compactness of generalized Volterra type integral operatorsGI(ϕ,g)f(z)=∫0zf′(ϕ(ξ))g(ξ)dξ andGV(ϕ,g)f(z)=∫0zf(ϕ(ξ))g(ξ)dξ, acting between large Bergman spaces Aωp and Aωq for 0<p,q≤∞. To prove our characterizations, which involve Berezin type integral transforms, we use the Littlewood-Paley formula of Constantin and Peláez and establish corresponding embedding theorems, which are also of independent interest. When ϕ(z)=z, our results for GV(ϕ,g) complement the descriptions of Pau and Peláez.

Essential norm of generalized integral type operator from QK(p,q) to Zygmund spaces

Let ? be an analytic self-map on D, n ? N and 1 ? H(D). We consider the essential norm of the generalized integral-type operator Cn ?,g : QK (p,q) ? Z? that is defined as follows (Cn ?,g f) (z) = ?z0 f(n)(?(?))1(?) d?, for all f ? QK (p,q). We give an estimate for the essential norm of the above operator.

Generalized Stević-Sharma type operators from derivative Hardy spaces into Zygmund-type spaces

&lt;abstract&gt;&lt;p&gt;Let $ u, v $ be two analytic functions on the open unit disk $ {\mathbb D} $ in the complex plane, $ \varphi $ an analytic self-map of $ {\mathbb D} $, and $ m, n $ nonnegative integers such that $ m &amp;lt; n $. In this paper, we consider the generalized Stević-Sharma type operator $ T_{u, v, \varphi}^{m, n}f(z) = u(z)f^{(m)}(\varphi(z))+v(z)f^{(n)}(\varphi(z)) $ acting from the derivative Hardy spaces into Zygmund-type spaces, and investigate its boundedness, essential norm and compactness.&lt;/p&gt;&lt;/abstract&gt;

Weighted composition operators from the Besov space into nth weighted type spaces

Abstract Let φ be an analytic self-map of the open unit disk 𝔻 in the complex plane ℂ and u be an analytic function on 𝔻. The weighted composition operator is defined on the space H(𝔻) of analytic functions on 𝔻 by u C φ f = u ⋅ ( f ∘ φ ) , f ∈ H ( D ) . $$u{C_\varphi }f = u \cdot \left( {f \circ \varphi } \right),\quad f \in H\left(D\right).$$ The boundedness and the compactness of weighted composition operators from the Besov space into nth weighted type spaces are given in this work.

Composition operators on Hardy-Smirnov spaces

We investigate composition operators CΦ on the Hardy-Smirnov space H2(Ω) induced by analytic self-maps Φ of an open simply connected proper subset Ω of the complex plane. When the Riemann map τ:U→Ω used to define the norm of H2(Ω) is a linear fractional transformation, we characterize the composition operators whose adjoints are composition operators. As applications of this fact, we provide a new proof for the adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez and we give a new approach to describe all Hermitian and unitary composition operators on H2(Ω). These descriptions are particular cases of the results obtained by Gunatillake-Jovovic-Smith in 2015 and Gunatillake in 2017. Additionally, if the coefficients of τ are real, we exhibit concrete examples of conjugations and describe the Hermitian and unitary composition operators which are complex symmetric with respect to specific conjugations on H2(Ω). We finish this paper showing that if Ω is unbounded and Φ is a non-automorphic self-map of Ω with a fixed point, then CΦ is never complex symmetric on H2(Ω).

Sum of generalized weighted composition operators between weighted spaces of analytic functions

Let [Formula: see text] be the space of analytic functions on the unit disc [Formula: see text] and let [Formula: see text] denote the set of analytic self-maps of [Formula: see text]. Let [Formula: see text] be such that [Formula: see text] and [Formula: see text]. We characterize the boundedness, compactness and completely continuous of the sum of generalized weighted composition operators [Formula: see text] between weighted Banach spaces of analytic functions [Formula: see text] and [Formula: see text] which unifies the study of products of composition operators, multiplication operators and differentiation operators. As applications, we obtain the boundedness and compactness of the generalized weighted composition operators [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are weighted Bloch-type (little Bloch-type) spaces. Also, new characterizations of the boundedness and compactness of the operators [Formula: see text] and [Formula: see text] are given. Examples of bounded, unbounded, compact and non-compact operators [Formula: see text] and [Formula: see text] are given to explain the role of inducing functions [Formula: see text], [Formula: see text] and the weights [Formula: see text], [Formula: see text] of the underlying weighted spaces.

Product-type operators on Banach spaces of analytic functions on the upper half-plane

Let , the upper half-plane in the complex plane . Let be the set of all analytic functions on Π and be the set of all analytic self-maps on Π. For , and , where , , define an operator In this paper, we characterize the boundedness and compactness of the operator on the Hardy, weighted-Bergman spaces on Π to weighted-type spaces on Π.

Isometries of the product of composition operators on weighted Bergman space

In this paper, the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on [Formula: see text] and [Formula: see text] the composition operator on [Formula: see text] induced by an analytic self-map on [Formula: see text] with fixed origin need not be of norm one. We have generalized the Schwartz’s [Composition operators on [Formula: see text], thesis, University of Toledo (1969)] well-known result on [Formula: see text] which characterizes the almost multiplicative operator on [Formula: see text]

Isometric multiplication operators and weighted composition operators of BMOA

Let u be an analytic function in the unit disk mathbb{D} and φ be an analytic self-map of mathbb{D}. We give characterizations of the symbols u and φ for which the multiplication operator M_{u} and the weighted composition operator M_{u,varphi } are isometries of BMOA.

Another look at the Landau theorem

Let \(f\) be the analytic self-map of the open unit disk \(\mathbf{D}\) in complex plane with \(f(0)=0\) and \(f'(0)\neq 0\). The classical Landau theorem shows that the image \(f(\mathbf{D})\) contains a schlicht disk. In this note, it is proved that the injectivity of \(f\) in the Landau theorem can be strengthened to be starlike. In particular, it provides a way to construct starlike functions from bounded analytic functions. Furthermore, we obtain a new version of the Landau theorem for vector-valued analytic functions from the perspective of modulus functions.

Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces

The operators D C Φ and C Φ D are defined by D C Φ f = f ∘ Φ ′ and C Φ D f = f ′ ∘ Φ where Φ is an analytic self-map of the unit disc and f is analytic in the disc. A characterization is provided for boundedness and compactness of the products of composition and differentiation from the spaces of fractional Cauchy transforms F α to the Bloch-type spaces B β , where α &gt; 0 and β &gt; 0 . In the case β &lt; 2 , the operator D C Φ : F α ⟶ B β is compact ⇔ D C Φ : F α ⟶ B β is bounded ⇔ Φ ′ ∈ B β , Φ Φ ′ ∈ B β and Φ ∞ &lt; 1 . For β &lt; 1 , C Φ D : F α ⟶ B β is compact ⇔ C Φ D : F α ⟶ B β is bounded ⇔ Φ ∈ B β and Φ ∞ &lt; 1 .

Level Sets of the Hyperbolic Derivative for Analytic Self-Maps of the Unit Disk

Let the function $$\varphi $$ be holomorphic in the unit disk $${\mathbb {D}}$$ of the complex plane $${\mathbb {C}}$$ and let $$\varphi ({\mathbb {D}})\subset {\mathbb {D}}$$ . We study the level sets and the critical points of the hyperbolic derivative of $$\varphi $$ , $$\begin{aligned} |D_{\varphi }(z)|:=\frac{(1-|z|^2)|\varphi '(z)|}{1-|\varphi (z)|^2}. \end{aligned}$$ In particular, we show how the Schwarzian derivative of $$\varphi $$ reveals the nature of the critical points.

Semigroups of composition operators on [formula omitted] spaces

In this paper we prove that no non-trivial semigroup (φt)t≥0 consisting of analytic self-maps of the unit disk D generates a strongly continuous composition semigroup on Qp for p>0; that is, [φt,Qp]⫋Qp, which answers a question asked by A.G. Siskakis in 1996. We also give an affirmative answer to question in [9] that Qp,0⫋[φt,Qp] for all p>0 if the Denjoy-Wolff point of (φt)t≥0 is on the boundary of D, where Qp,0 is the closure of all polynomials in Qp space. Moreover, for a semigroup (φt)t≥0 with the Denjoy-Wolff point in D, we give a sufficient and necessary condition such that Qp,0⫋[φt,Qp] in terms of the infinitesimal generator of (φt)t≥0 for all p>0. As an application, a new characterization of Qp,0 in terms of semigroup (φt)t≥0 acting on Qp is given, which gives a generalization of the theorem of D. Sarason for VMOA.

Complex Symmetry of Invertible Composition Operators on Weighted Bergman Spaces

In this article, we study the complex symmetry of compositions operators $C_{\phi}f=f\circ \phi$ induced on weighted Bergman spaces $A^2_{\beta}(\mathbb{D}),\ \beta\geq -1,$ by analytic self-maps of the unit disk. One of ours main results shows that $\phi$ has a fixed point in $\mathbb{D}$ whenever $C_{\phi}$ is complex symmetric. Our works establishes a strong relation between complex symmetry and cyclicity. By assuming $\beta\in \mathbb{N}$ and $\phi$ is an elliptic automorphism of $\mathbb{D}$ which not a rotation, we show that $C_{\phi}$ is not complex symmetric whenever $\phi$ has order greater than $2(3+\beta).$