On Algebras Generated by Composition Operators

Let $$\mathbb {D}$$ be the open unit disk in the complex plane $$\mathbb {C}$$, and let $$\varphi $$ be a holomorphic function from disk $$\mathbb {D}$$ into $$\mathbb {D}$$. We study the composition operator $$C_{\varphi }$$ on the variable exponent Bergman space in unit disk, and prove that this operator is bounded on $$A^{p(\cdot )}(\mathbb {D})$$. We give a sufficient condition for the compactness of this operator on $$A^{p(\cdot )}(\mathbb {D})$$ as well.

Let 𝔻 be the open unit disk in the complex plane ℂ and \(dA(z) = \frac {1}{\pi }dx dy\), the normalized area measure on 𝔻. Let \({L_{a}^{2}}(\mathbb {D}, dA)\) be the Bergman space consisting of analytic functions on 𝔻 that are also in \(L^{2}(\mathbb {D}, dA)\). Let \(\mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) be the set of all bounded linear operators from the Hilbert space \({L_{a}^{2}}(\mathbb {D})\) into itself. For \(T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\), let \(\widetilde {T}\) denote the Berezin transform of T. In this paper, we find conditions on positive operators \(S, T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) such that \(\widetilde {S}(z)\geq \widetilde {T}(z)\) for all z ∈ 𝔻.

The boundedness and the compactness of the two integral operators;, where is an analytic function on the open unit disk in the complex plane, on the Zygmund space are studied.

Let \({\mathbb{D}}\) be the open unit disk in the complex plane \({\mathbb{C}}\) and \({H(\mathbb{D})}\) the class of all analytic functions on \({\mathbb{D}}\). Let \({\phi}\) be an analytic self-map of \({\mathbb{D}}\) and \({u \in H(\mathbb{D})}\). As a class of new space, we first study the properties of the logarithmic Bergman-type space. We also use Young’s function to define another new space as a generalization of Zygmund space, called the Zygmund–Orlicz space. We study some of its properties and show that the Zygmund–Orlicz space is isometrically equal to certain Zygmund-type space for a very special weight. This allows us to define the little Zygmund–Orlicz space which is a closed subspace of the Zygmund–Orlicz space. As some applications of properties obtained, we characterize the boundedness and compactness of the product-type operators \({D^{n} M_{u}C_{\phi}}\), \({D^{n}C_{\phi}M_{u}}\), \({C_\phi D^n M_u}\), \({M_uD^nC_{\phi}}\), \({M_{u}C_{\phi}D^{n}}\), and \({C_{\phi}M_{u}D^{n}}\) from the logarithmic Bergman-type space to the Zygmund–Orlicz space and the little Zygmund–Orlicz space.

Let D be the open unit disk in the complex plane, and let C be its boundary, the unit circle.If x £ C, then by an arc at x we mean a simple arc y with one end point at x such that y-{x}£ö.If/is a function mapping D into some metric space M, then the set of curvilinear convergence of/is defined to be {x e C: there exists an arc y at x and there exists a point p e M such that f(z) ->p as z -»■ x along y}.If ^ is a function whose domain is a subset F of the set of curvilinear convergence of/ then <f> is called a boundary function for/if, and only if, for each x e E there exists an arc y at x such that/(z) -*■ <f>(x) as z -> x along y.Let 5 be another metric space.We shall say that a function <f> is of Baire class ^ 1(5, M) if (i) domain <£ = S, (ii) range ^ £ M, and(iii) there exists a sequence {<£"} of continuous functions, each mapping S into M, such that <f>n^>-<p pointwise on 5. We shall say that (f> is of honorary Baire class ^ 2(S, M) if (i) domain <f> = S, (ii) range ^£M, and (iii) there exists a countable set TVs 5 and there exists a function y!r of Baire class ^ 1(5, M) such that <f>(x) = tp(x) for every xe S-N.It is known that if/is a continuous function mapping F into the Riemann sphere, then the set of curvilinear convergence off is of type Fa6, and any boundary function for/is of honorary Baire class ^2(C, Riemann sphere).(See [3], [4], [5], [6],[9].) J. E. McMillan [6] posed the following problem.If A is a given set in C of type Fao, and if </> is a function of honorary Baire class Ú2(A, Riemann sphere), does there always exist a continuous function / mapping D into the Riemann sphere such that A is the set of curvilinear convergence of/ and <f> is a boundary function for /?The purpose of this paper is to give an affirmative answer to McMillan's question.However, the corresponding question for real-valued functions remains open.(See Problems 1 and 2 at the end of this paper.)In proving our result, we first give a proof under the assumption that <j> is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere.As we shall indicate in an appendix, the same device can be

The main purpose of this paper is to obtain an upper bound for the second Hankel determinant for functions belonging to a subclass of bi-univalent functions in the open unit disk in the complex plane. Furthermore, the presented results in this work improve or generalize the recent works of other authors.

In this short note, the norm of Berezin transform, acting on $L^{p}(\mathbb{B}_{n})$, is determined to be \[\bigl\|B_{\mathbb{B} _{n}}:L^{p}(\mathbb{B}_{n})\to L^{p}(\mathbb{B}_{n})\bigr\|=\frac{1}{p}\prod_{k=1}^{n}\biggl(1+\frac{1}{kp}\biggr)\frac{\pi}{\sin(\pi/p)}.\] This extends a result of Dostanić (J. Anal. Math. 104 (2008) 13–23) to several complex variables.

We study the following two integral operators equation image where g is an analytic function on the open unit disk in the complex plane. The boundedness and compactness of these two operators between the α ‐Bloch space Bα and the Besov space are discussed in this paper (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Abstract. AsequenceofcompositionoperatorsonHardyspaceiscon-sidered. Weprovethat,bynumericalrangeproperties,itisSOT-conver-gencebutnotconverge. 1. introductionLet ϕ be a holomorphic self-map of the unit disc U := {z ∈ C : |z| < 1}.The function ϕ induces the composition operator C ϕ , defined on the space ofholomorphic functions on Uby C ϕ f = f ◦ ϕ. The restriction of C ϕ to variousBanach spaces of holomorphic functions on U has been an active subject ofresearch for more than three decades and it will continue to be for decadesto come (see [11], [12] and [4]). Let H 2 denote the Hardy space of analyticfunctions on the open unit disc with square summable Taylor coefficients. Inrecent yearsthe study ofcomposition operatorson the Hardy space has receivedconsiderable attention.In this paper we consider the numerical range of elliptic composition opera-tors on H 2 . The numerical range of a bounded linear operator A on a Hilbertspace H is the set of complex numbersW(A) := {hAx,xi : x ∈ H,kxk = 1},where h·,· denotes the inner product in H.In [9] V. Matache determined the shape W(C

Let be the unit disk in the complex plane. We define to be the little Bloch space of functions f analytic in which satisfy lim|z|→1(1 - |z|2)|f'(z)| = 0. If is analytic then the composition operator C φ : f ↦ f ∘ φ is a continuous operator that maps into itself. In this paper, we show that the compactness of C φ , as an operator on , can be modelled geometrically by its principal eigenfunction. In particular, under certain necessary conditions, we relate the compactness of C φ to the geometry of , where σ satisfies Schöder's functional equation σ ∘ φ = φ'(0)σ.2000 Mathematics Subject Classification: Primary 30D05; 47B33 Secondary 30D45.

Consider an open unit disk D = z ∈ ℂ : z &lt; 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.

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).

We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $H^{p}$, $1\leq p<\infty$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain.

Let \( A^2 (\mathbb{D}) \) be the Bergman space over the open unit disk \( \mathbb{D} \) in the complex plane. Korenblum conjectured that there is an absolute constant \( c \in (0,1) \), such that whenever \( |f(z)| \leq |g(z)| (f, g \in A^2 (\mathbb{D})) \) in the annulus \( c < |z| < 1 \), then \( ||f|| \leq ||g|| \). In 1999 Hayman proved Korenblum’s conjecture. But the sharp value of c (we use γ to denote this sharp value and call it Korenblum’s constant) is still unknown. In this paper we give an upper bound on γ, that is, γ < 0.685086, which improves an earlier result of the author.

Let ϕ be an analytic self-map and u be a fixed analytic function on the open unit disk 𝔻 in the complex plane ℂ. The following integral-type operators have recently attracted a lot of attention. In this article, we characterize the boundedness and compactness of operators I g and J g from weighted Bloch spaces into Bergman-type spaces.