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Given a doubling weight ω on the unit disk D , let Aωp be the space of all the holomorphic functions f, where ∥ f ∥ A p := (∫ D ∣ f (z) ∣ p (z) d A (z)) 1 / p < ∞. We completely characterize the topological connectedness of the set of composition operators on Aωp. As an application, we construct an interesting example which reveals that two composition operators on Aαp in the same path component may fail to have a compact difference and give a negative answer to the Shapiro-Sundberg question in the (standard) weighted Bergman space. In addition, we completely describe the central compactness of any finite linear combinations of composition operators on Aωp in three terms: a Julia-Carathéodory-type function-theoretic characterization, a power-type characterization, and a Carleson-type measure-theoretic characterization. [ABSTRACT FROM AUTHOR]

In the paper 'Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363', Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with $ c_0\ge 1 $ c 0 ≥ 1. But the sufficient conditions for the other case $ c_0=0 $ c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case $ c_0=0 $ c 0 = 0. Moreover, we also obtain the compact characterizations of composition operators with $ c_0\geq 1 $ c 0 ≥ 1. [ABSTRACT FROM AUTHOR]

We obtain a new boundedness criterion for the difference of two composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < q < p and α > - 1 . As a consequence, we provide a direct proof that such a bounded difference operator is necessarily compact. We also characterize compact differences of composition operators from A α p into A β q explicitly for 0 < p ≤ q and α , β > - 1 . [ABSTRACT FROM AUTHOR]

We study composition operators between the weighted Bergman space of the polydisk and the weighted Bergman space of the unit disk. We find conditions on the symbols to guarantee the boundedness and compactness of the composition operator. We shall also find a necessary and sufficient condition for the composition operator to be Hilbert–Schmidt. [ABSTRACT FROM AUTHOR]

It is known that the norm of the Hilbert matrix operator on weighted Bergman spaces A α p was conjectured by Karapetrović to be π sin ⁡ (α + 2) π p when α > − 1 and p > α + 2. The conjecture has been confirmed by Božin and Karapetrović in the case α = 0. In this paper we prove the conjecture for the cases both α = 1 and 0 < α ≤ 1 47. Moreover, we also show that the conjecture is valid when − 1 < α < 0 and p ≥ 2 (α + 2). [ABSTRACT FROM AUTHOR]

Motivated by the recent work of Galindo et al. (J. Funct. Anal. 265, 629–643, 2013), in this paper, we give an elegant compactness criterion for any finite linear combination of composition operators on the weighted Bergman space in terms of power type characterization. More precisely, let T be any finite linear combination of composition operators, then T is compact on A α p (D) if and only if lim n → ∞ ∥ T z n ∥ A − γ ∥ z n ∥ A − γ = 0 , which reveals that the compactness of T on the weighted Bergman space A α p (D) and Korenblum space A−γ(D) are equivalent. [ABSTRACT FROM AUTHOR]

Abstract Closures of weighted Bergman spaces in Bloch type spaces are investigated in the paper. Moreover, the boundedness and compactness of the product of composition and differentiation operators from Bloch type spaces to closures of weighted Bergman spaces spaces in Bloch type spaces are characterized.

To overcome the unboundedness of the half-plane, we use Khinchine's inequality and atom decomposition techniques to provide joint Carleson measure characterizations when the difference of composition operators is bounded or compact from standard weighted Bergman spaces to Lebesgue spaces over the half-plane for all index choices. For applications, we obtain direct analytic characterizations of the bounded and compact differences of composition operators on such spaces. This paper concludes with a joint Carleson measure characterization when the difference of composition operators is Hilbert-Schmidt. [ABSTRACT FROM AUTHOR]

Let φ be an analytic self-map of the open unit disk D. We study the complex symmetry of composition operators Cφ on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D that is not an elliptic automorphism, we establish that if Cφ is complex symmetric, then either φ (0) = 0 or φ is linear. In the case of weighted Bergman spaces A2α, we find the non-automorphic linear fractional symbols φ such that Cφ is complex symmetric. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the difference of composition-differentiation operators D φ − D ψ acting from Hardy spaces H p to weighted Bergman spaces A α q are completely characterized for all 0 < p , q < ∞. [ABSTRACT FROM AUTHOR]

Closures of weighted Bergman spaces in Bloch type spaces are investigated in the paper. Moreover, the boundedness and compactness of the product of composition and differentiation operators from Bloch type spaces to closures of weighted Bergman spaces spaces in Bloch type spaces are characterized. [ABSTRACT FROM AUTHOR]

We extend known results on the spectra of composition operators to the weighted Bergman spaces. Our results include a study of the essential spectral radius, a determination of the spectrum when the symbol of the composition operator is univalent and non-automorphic with a fixed point in the disk, and an affirmative answer to a conjecture of MacCluer and Saxe. [ABSTRACT FROM AUTHOR]

Abstract We investigate the topological connectedness of the set of composition operators on the weighted Bergman space A α 2 under the uniform operator topology. One of our main results is to first find a necessary and sufficient condition for two composition operators on A α 2 to be linearly connected in the case of one variable. As a corollary, we show that two composition operators on A α 2 with compact difference must lie in the same path component. [ABSTRACT FROM AUTHOR]

In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of causal weighted composition operators on these spaces are related to each other and use it to show that an (unweighted) composition operatorCφ is bounded on a Zen space if and only if φ has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces. [ABSTRACT FROM AUTHOR]

An analytic self-map ϕ of the open unit disk 픻 in the complex plane induces the so-called composition operator Cϕ : H(픻) → H(픻), f ↦ f ∘ ϕ, where H(픻) denotes the set of all analytic functions on 픻. Motivated by [5] we analyze under which conditions on the weight v all composition operators Cϕ acting between the weighted Bergman space and the weighted Banach space of holomorphic functions both generated by v are bounded. [ABSTRACT FROM AUTHOR]

As is well known on the weighted Bergman spaces over the unit disk, compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every “bad” boundary point, which makes each composition operator in the difference fail to be compact. Recently, the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators. In this paper, we obtain a complete characterization for compact double differences formed by four composition operators. Applying our characterization, we easily recover known results on linear combinations of two or three composition operators. As another application, we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, our characterization also shows that double difference cancellation may occur in the global sense only, and that genuine double difference cancellation is not possible in a certain local sense. [ABSTRACT FROM AUTHOR]

Let $$\varphi $$ be an arbitrary linear-fractional self-map of the unit disk $${\mathbb {D}}$$ and consider the composition operator $$C_{-1, \varphi }$$ and the Toeplitz operator $$T_{-1,z}$$ on the Hardy space $$H^2$$ and the corresponding operators $$C_{\alpha , \varphi }$$ and $$T_{\alpha , z}$$ on the weighted Bergman spaces $$A^2_{\alpha }$$ for $$\alpha >-1$$ . We prove that the unital C $$^*$$ -algebra $$C^*(T_{\alpha , z}, C_{\alpha , \varphi })$$ generated by $$T_{\alpha , z}$$ and $$C_{\alpha , \varphi }$$ is unitarily equivalent to $$C^*(T_{-1, z}, C_{-1, \varphi }),$$ which extends a known result for automorphism-induced composition operators. For maps $$\varphi $$ that are not automorphisms of $${\mathbb {D}}$$ , we show that $$C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })$$ is unitarily equivalent to $$C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})$$ , where $${\mathcal {K}}_{\alpha }$$ and $${\mathcal {K}}_{-1}$$ denote the ideals of compact operators on $$A^2_{\alpha }$$ and $$H^2$$ , respectively, and apply existing structure theorems for $$C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})/{\mathcal {K}}_{-1}$$ to describe the structure of $$C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })/\mathcal {K_{\alpha }}$$ , up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps $$\varphi $$ that fix a point on the unit circle. [ABSTRACT FROM AUTHOR]

Let be the composition operator induced by a holomorphic self-map φ of the open complex unit disk. In this paper, a necessary and sufficient condition for the boundedness of from one weighted Bergman space of logarithmic weight into another is described in terms of a growth condition of a generalized counting function for φ. We make use of a new integral representation of a modified counting function which depends on log-convexity of the weight function as well as some estimates for the norm of the weighted Bergman space. [ABSTRACT FROM AUTHOR]

The compact differences of composition operators acting on the weighted L-Bergman space over the unit disk is characterized by the angular derivative cancellation property and due to Moorhouse. In this paper we extend Moorhouse's characterization, as well as some related results, to the ball and, at the same time, to the weighted L-Bergman space for the full range of p. [ABSTRACT FROM AUTHOR]

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are $${(1 - \overline{w}z)^{-\kappa}}$$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces. [ABSTRACT FROM AUTHOR]

For linear-fractional self-maps ϕ and ψ of the unit disc 𝔻, where at least one of ϕ and ψ is a non-automorphism, we show that the commutatoris non-trivially compact on the weighted Bergman spaceif and only if either ϕ and ψ are both parabolic or ϕ and ψ are both hyperbolic, with associated conclusions about their fixed points in each case. In the automorphism case, we show that this commutator is compact if and only if both ϕ and ψ are rotations. [ABSTRACT FROM PUBLISHER]

Moorhouse characterized compact differences of composition operators acting on a weighted Bergman space over the unit disk of the complex plane. She also found a sufficient condition for a single composition operator to be a compact perturbation of the sum of given finitely many composition operators and studied the role of second order data in determining compact differences. In this paper, based on the characterizations due to Stessin and Zhu, of boundedness and compactness of composition operators acting from a weighted Bergman space into another, we obtain the polydisk analogues of Moorhouse's results through a different approach in main steps. In addition we find a necessary coefficient relation for compact combinations which was first noticed on the disk by Kriete and Moorhouse. [ABSTRACT FROM AUTHOR]

Abstract: We show that the approximation numbers of a compact composition operator on the Hardy space or on the weighted Bergman spaces of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they cannot decay more rapidly than exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bound and explicit an example. [Copyright &y& Elsevier]

We estimate the essential norm of a composition operator acting on the Hardy space H and the weighted Bergman spaces $${A_{\alpha}^{p}}$$ on the unit ball. In passing, we recover (and somehow simplify the proof of) parts of the recent article by Demazeux, dealing with the same question for H of the unit disc. We also estimate the essential norm of a composition operator acting on $${A_{\alpha}^{p}}$$ in terms of the angular derivatives of $${\phi}$$, under a mild condition on $${\phi}$$. [ABSTRACT FROM AUTHOR]

For analytic self-maps φ of the unit disk, we show that recently given conditions that are both necessary and sufficient for the composition operator C to be closed-range on the classical Bergman space $${\mathbb{A}^2}$$ are actually necessary and sufficient in the general setting of the weighted Bergman spaces $${\mathbb{A}^p_{\alpha}}$$ , for 1 ≤ p < ∞ and weights of the form $${(1 - |z|^2)^{\alpha}}$$ ; where α > −1. Along the way, we show how work done in a paper of J. Akeroyd, P. Ghatage and M. Tjani can be used to improve upon the most definitive of these conditions, and we give applications. [ABSTRACT FROM AUTHOR]

We use induction and interpolation techniques to prove that a composition operator induced by a map ϕ is bounded on the weighted Bergman space $\mathcal{A}^2_\alpha(\mathbb{H})$ of the right half-plane if and only if ϕ fixes the point at ∞ non-tangentially and if it has a finite angular derivative λ there. We further prove that in this case the norm, the essential norm and the spectral radius of the operator are all equal and are given by λ(2+α)/2. [ABSTRACT FROM PUBLISHER]

Abstract: The boundedness of the composition operator from the weighted Bergman space to, recently introduced by this author, the nth weighted-type space on the upper half-plane are characterized here. [Copyright &y& Elsevier]

The boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces are discussed. [ABSTRACT FROM AUTHOR]

The boundedness of products of differentiation operators and composition operators between the weighted Bergman space and the Bloch type space are discussed in this paper. [ABSTRACT FROM AUTHOR]

Abstract: We observe that a formula for the adjoint of a composition operator, known only for special symbols in some spaces of analytic functions, actually holds for every admissible symbol and in any Hilbert space of analytic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form. [Copyright &y& Elsevier]

The characterization of path components in the space of composition operators acting in various settings has been a long-standing open problem. Recently Dai has obtained a characterization of when two composition operators acting on the weighted Hilbert-Bergman space on the unit disk are linearly connected, i.e., they are joined by a continuous "line segment" of composition operators induced by convex combinations of the maps inducing the two given composition operators. In this paper we consider composition operators acting on the weighted Bergman spaces over the half-plane. Since not all composition operators are bounded in this setting, we introduce a metric induced by the operator norm and study when (possibly unbounded) composition operators are linearly connected in the resulting metric space. We obtain necessary conditions that under a natural additional assumption are also sufficient. We also study the problem of when a composition operator is isolated. Complete results are obtained for composition operators induced by linear fractional self-maps of the half-plane. We show that the only such composition operators that are isolated are those induced by automorphisms of the half-plane. We also characterize when composition operators induced by linear fractional self-maps belong to the same path component. The characterization demonstrates that composition operators in the same path component may have inducing maps with different behavior at infinity. In contrast, when the setting is the disk the corresponding boundary behavior of the inducing maps must match. [ABSTRACT FROM AUTHOR]

Motivated by the question of Shapiro and Sundberg raised in 1990, study on linear combinations of composition operators has been a topic of growing interest. In this paper, we completely characterize the compactness of any finite linear combination of composition operators with general symbols on the weighted Bergman spaces in two classical terms: one is a function theoretic characterization of Julia-Carathéodory type and the other is a measure theoretic characterization of Carleson type. Our approach is completely different from what has been known so far. [ABSTRACT FROM AUTHOR]

An analytic self-map φ of the open unit disk D in the complex plane induces the so-called composition operator Cφ : H(D) → H(D), f 7→ f ◦ φ, where H(D) denotes the set of all analytic functions on D. Motivated by [5] we analyze under which conditions on the weight v all composition operators Cφ acting between the weighted Bergman space and the weighted Banach space of holomorphic functions both generated by v are bounded. [ABSTRACT FROM AUTHOR]