Analytic Self-map Research Articles

Composition Operators on Generalized Weighted Nevanlinna Class

Let $\varphi$ be an analytic self-map of open unit disk $\mathbb{D}$. The operator given by $(C_{\varphi}f)(z)=f(\varphi(z))$, for $z \in \mathbb{D}$ and $f$ analytic on $\mathbb{D}$ is called a composition operator. Let $\omega$ be a weight function such that $\omega\in L^{1}(\mathbb{D}, dA)$. The space we consider is a generalized weighted Nevanlinna class $\mathcal{N}_{\omega}$, which consists of all analytic functions $f$ on $\mathbb{D}$ such that $\displaystyle \|f\|_{\omega}=\int_{\mathbb{D}}\log^{+}(|f(z)|) \omega(z) dA(z)$ is finite; that is, $\mathcal{N}_{\omega}$ is the space of all analytic functions belong to $L_{\log^+}(\mathbb{D}, \omega dA)$. In this paper we investigate, in terms of function-theoretic, composition operators on the space $\mathcal{N}_{\omega}$. We give sufficient conditions for the boundedness and compactness of these composition operators.

Topological Structure of the Space of Weighted Composition Operators Between Different Hardy Spaces

We consider properties related to weighted composition operators boundedly acting from the classical Hardy space H p to H q for $${1 \leq q < p < \infty}$$ . Especially, we shall completely determine path connected components in the set of weighted composition operators and explicitly characterize by function-theoretic properties of analytic self-maps.

Boundary vs. interior conditions associated with weighted composition operators

Abstract Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk $$\mathbb{D}$$, we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior $$\mathbb{D}$$ and on the boundary $$\partial \mathbb{D}$$ respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.

Composition operators for which and commute

We define . In this paper, we characterize composition operators and their adjoints which belong to , where the maps are linear fractional selfmaps of the open unit disk into itself. If is an automorphism of or , then the case for is precisely when it is normal. When , we also prove that if , then either or , which implies that the only binormal composition operators with and are normal. Moreover, we show that if and is not normal, then implies that and is neither real nor purely imaginary, while ensures that and is real. Finally, we study composition operators in where is an analytic selfmap into . In particular, this operator has the single-valued extension property.

Inequalities on Hardy and higher-power weighted Bergman spaces of composition operators

Bounded composition operators are usually induced by analytic self-maps of the open unit disk acting on the Hardy space H 2 and on the higher-power weighted Bergman spaces L 2 where e = ( + 1) 2 1. An inequality for the relationship between the norms of the corresponding composition operators defined on these spaces is considered.

Generalized Composition Operators fromℬμSpaces toQK,ω(p,q)Spaces

Let0&lt;p&lt;∞, let-2&lt;q&lt;∞, and letφbe an analytic self-map of𝔻andg∈H(𝔻). The boundedness and compactness of generalized composition operators(Cφgf)(z)=∫0z‍f'(φ(ξ))g(ξ)dξ, z∈𝔻, f∈H(𝔻), fromℬμ(ℬμ,0) spaces toQK,ω(p,q)spaces are investigated.

Differences of generalized composition operators between Bloch type spaces

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.

A NEW ESSENTIAL NORM ESTIMATE OF COMPOSITION OPERATORS FROM α-BLOCH SPACES INTO μ-BLOCH SPACES

Let μ be any weight function defined on the unit disk 𝔻 and let ϕ be an analytic self-map of 𝔻. In this paper, we show that the essential norm of composition operator Cϕmapping from the α-Bloch space, with α &gt; 0, to μ-Bloch space [Formula: see text] is comparable to [Formula: see text] where, for a ∈ 𝔻, σais a certain special function in α-Bloch space. As a consequence of our estimate, we extend recent results, about the compactness of composition operators, due to Tjani in [Compact composition operators on Besov spaces, Trans. Amer. Math. Soc.355(11) (2003) 4683–4698] and Malavé Ramírez and Ramos-Fernández in [On a criterion for continuity and compactness of composition operators acting on α-Bloch spaces, C. R. Math. Acad. Sci. Paris351 (2013) 23–26, http://dx.doi.org/10.1016/j.crma.2012.11.013 ].

Compact composition operators on Bergman-Orlicz spaces

We construct an analytic self-map ' of the unit disk and an Orlicz functionfor which the composition operator of symbol ' is compact on the Hardy-Orlicz space H � , but not on the Bergman-Orlicz space B � . For that, we first prove a Carleson embedding theorem, and then characterize the compact- ness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2. Mathematics Subject Classification. Primary: 47B33 - Secondary: 30D50; 30D55; 46E15

Weighted Composition Operators Between Zygmund Type Spaces and Their Essential Norms

Let \({u \in \mathcal{H}(\mathbb{D})}\) and φ be an analytic self-map of \({\mathbb{D}}\). We estimate the essential norms of weighted composition operators uCφ acting on Zygmund type spaces in terms of u, φ, their derivatives and the n-th power φn of φ. Moreover, we give similar characterizations for boundedness of uCφ between Zygmund type spaces.

Boundary asymptotic expansions of analytic self-maps of the unit disk

A characterization of analytic self-maps of the open unit disk in terms of their Taylor coefficients is due to I. Schur. We present a boundary analog of this result.

Difference of composition operators on Hardy space

(Communicated by G. Sinnamon) Abstract. Suppose ϕ is an analytic self-map of open unit disk D and w is an analytic func- tion on D. Then a weighted composition operator induced by ϕ with weight w is given by (Ww,ϕ f)(z )= w(z)f(ϕ(z)) ,f orz ∈ D and f analytic on D.W efind a sufficient condition un- der which two composition operators lie in the same path component of C(H 2 ) ,a nd wefind as ufficient condition for the difference of such operators to be compact on H 2 (D) .T hen we provide another example that answers a question raised by Shapiro and Sundberg (18 )n ega- tively. Moreover, we characterize the Hilbert-Schmidt difference of two composition operators on H2(D).

Hilbert-Schmidt differences of composition operators between the weighted Bergman spaces on the unit ball

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

Products of composition and n-th differentiation operators from α-bloch space to Qp space

Let ? be an analytic self-map of the open unit disk D on the complex plane and ? &gt; 0, p ? 0, n ? N. In this paper, the boundedness and compactness of the products of composition operators and nth differentiation operators C?Dn from a-Bloch space B? and B?0 to Qp space are investigated.

A New Essential Norm Estimate of Composition Operators from Weighted Bloch Space into -Bloch Spaces

Let be any weight function defined on the unit disk and let be an analytic self-map of . In the present paper, we show that the essential norm of composition operator mapping from the weighted Bloch space to -Bloch space is comparable to where for , is a certain special function in the weighted Bloch space. As a consequence of our estimate, we extend the results about the compactness of composition operators due to Tjani (2003).

A note on composition operators in a half-plane

We provide conditions for composition operators on the Hardy space of the upper half-plane to have a closed range. We also show that, unlike the situation of the unit disk, the operator of composition with an analytic self-map Φ of the upper half-plane can be similar to an isometry even when Φ is far from being an inner function.

Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature

A classical result due to Blaschke states that for every analytic self-map f of the open unit disc of the complex plane there exists a Blaschke product B such that the zero sets of f and B agree. Indeed, a sequence is the zero set of an analytic self-map of the open unit disc if and only if it satisfies the simple geometric condition known as the Blaschke condition. In contrast, the critical sets of analytic self-maps of the open unit disc have not been completely described yet. In this paper, we show that for every analytic self-map f of the open unit disc there is even an indestructible Blaschke product B such that the critical sets of f and B coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighted Bergman space as well as to the Berger–Nirenberg problem from differential geometry. By solving the Berger–Nirenberg problem in a special case, we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space 𝒜12.

SUMS OF WEIGHTED COMPOSITION OPERATORS ON COP

AbstractLet COP =0∩H∞, where0is the little Bloch space on the open unit disk, andA() be the disk algebra on. For non-zero functionsu1,u2,. . .,uN∈A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕNsatisfying ϕj∈A() and ∥ϕj∥∞=1 for everyj, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕjmaps COP intoA().

Factorizations of analytic self-maps of the upper half-plane

We extend a factorization due to Kreuon to arbitrary analytic func- tions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has fewer zeros and singularities than the original one. The reult is used to construct functions with given zeros and poles on the real line.

Infinitesimal Carleson Property for Weighted Measures Induced by Analytic Self-Maps of the Unit Disk

We prove that, for every \(\alpha > -1\), the pull-back measure \(\varphi ({\mathcal A }_\alpha )\) of the measure \(d{\mathcal A }_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\mathcal A } (z)\), where \({\mathcal A }\) is the normalized area measure on the unit disk \(\mathbb D \), by every analytic self-map \(\varphi :\mathbb D \rightarrow \mathbb D \) is not only an \((\alpha \,{+}\, 2)\)-Carleson measure, but that the measure of the Carleson windows of size \(\varepsilon h\) is controlled by \(\varepsilon ^{\alpha + 2}\) times the measure of the corresponding window of size \(h\). This means that the property of being an \((\alpha + 2)\)-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman–Orlicz spaces.