In the setting of the Hilbert-Bergman space over the ball, Dai obtained a characterization of linear connection in the space of bounded composition operators. In this paper we improve Dai's results by considering the space of all (not necessarily bounded) composition operators and by working on the general standard weighted Bergman spaces. We also provide some sufficient conditions for (non-)isolated composition operators, characterize isolated linear fractional composition operators, and exhibit concrete examples negating the ball analogue of the Shapiro-Sundb erg question. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We introduce the Fejér-Riesz composition operator F_φ induced by a holomorphic self-map φ of the unit disk D , and defined on the space of holomorphic functions on D by F_φ f = χ _(-1,1)f∘φ . Denote by A^p_α the standard weighted Bergman space of holomorphic functions on D and denote the Hardy space H^p by A^p_-1 . We study the operators F_φ : A^p_α→ L^p(m_α ) , where m_α is the weighted Lebesgue measure dm_α (x) = (1-x^2)^α +1dx , x∈ (-1,1) . For 0

We study semigroups of composition operators acting on the Besov spaces ℬ_p , where they exhibit some new behaviors relative to many classical spaces. Often for a Banach space X of analytic functions on the unit disk, the maximal closed space of strong continuity, [φ _t, X] , exists for every semigroup {φ _t} of analytic self-maps of the disk, and the question whether [φ _t, X] equals X itself has an answer independent of {φ _t} . Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and H^∞ . For the disk algebra A, [φ _t, A] = A precisely when {φ _t}⊂ A . For ℬ_p with p ≥ 2 , every {φ _t}⊂ℬ^p and always [φ _t, ℬ_p] = ℬ_p , but this fails when 1< p < 2 . We give an example where {φ _t}⊂ℬ_p and yet the induced composition operators {C_t} are not bounded on ℬ_p and we do not know if [φ _t,ℬ_p] exists. If it does exist, it cannot be equal to ℬ_p . Under the hypothesis that there is a uniform bound for the operator norms of the {C_t} , 0 ≤ t ≤ 1 , we characterize the semigroups {φ _t} such that [φ _t, ℬ_p] = ℬ_p .

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.

Let H={z∈C:Imz>0} be the upper half plane, and denote by Lp(R), 1≤p<∞, the usual Lebesgue space of functions on the real line R. We define two “composition operators” acting on Lp(R) induced by a Borel function φ:R→H‾, by first taking either the Poisson or Borel extension of f∈Lp(R) to a function on H‾, then composing with φ and taking vertical limits. Classical composition operators, induced by holomorphic functions and acting on the Hardy spaces Hp(H) of holomorphic functions, correspond to a special case. Our main results provide characterizations of when the operators we introduce are bounded or compact on Lp(R), 1≤p<∞. The characterization for the case 1<p<∞ is independent of p and the same for the Poisson and the Borel extensions. The case p=1 is quite different.

In the setting of the Hardy spaces or the standard weighted Bergman spaces over the unit ball in \({\mathbf {C}}^n\), linear fractional composition operators are known to behave quite rigidly in the sense that they cannot form any nontrivial compact differences or, more generally, linear combinations. In this paper, in the setting of the standard weighted Bergman spaces over the half-plane, we completely characterize bounded/compact differences of linear fractional composition operators. Our characterization reveals that a linear fractional composition operator can possibly form a compact difference, which is a new half-plane phenomenon due to the half-plane not being bounded. Also, we obtain necessary conditions and sufficient conditions for a linear combination to be bounded/compact. As a consequence, when the weights and exponents of the weighted Bergman spaces are restricted to a certain range, we obtain a characterization for a linear combination to be bounded/compact. Applying our results, we provide an example showing a double difference cancellation phenomenon for linear combinations of three linear fractional composition operators, which is yet another half-plane phenomenon.

We obtain a necessary and sufficient condition for the operator of integration to be bounded on H∞ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions.

We study [φt,X], the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup {φt}t≥0 of analytic self-maps of the unit disk, when X is BMOA, H∞ or the disk algebra. In particular, we show that [φt,BMOA]≠BMOA for all nontrivial semigroups. We also prove, for every semigroup {φt}t≥0, that limt→0+⁡φt(z)=z not just pointwise, but in H∞ norm. This provides a unified proof of known results about [φt,X] when X∈{Hp,Ap,B0,VMOA}.

Let S be the unit sphere and B the unit ball in Cn, and denote by L1(S) the usual Lebesgue space of integrable functions on S. We define four “composition operators” acting on L1(S) and associated with a Borel function φ:S→B¯, by first taking one of four natural extensions of f∈L1(S) to a function on B¯, then composing with φ and taking radial limits. Classical composition operators acting on Hardy spaces of holomorphic functions correspond to a special case. Our main results provide characterizations of when the operators we introduce are bounded or compact on Lt(S), 1≤t<∞. Dependence on t and relations between the characterizations for the different operators are also studied.

Let Omega be an open simply connected proper subset of the complex plane. We identify, up to isomorphism, which groups are possible for the group of unitary composition operators of a Hardy-Smirnov space defined on Omega. We also study the relationship between the geometry of Omega and the corresponding group.

The operator that takes the function f to ψf∘φ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on H2, the Hardy–Hilbert space of the unit disc. We consider the case where φ is a rotation of the unit disc and identify a class of convexoid operators. In the case of isometric weighted composition operators we give a complete classification of their numerical ranges. We also consider the inclusion of zero in the interior of the numerical range.

Let f and g be analytic on the unit disk \({\mathbb{D}}\). The integral operator T g is defined by \({ T_g f(z) = \int_0^z f(t)g'(t) \,dt, z \in \mathbb{D}}\). The problem considered is characterizing those symbols g for which T g acting on H ∞, the space of bounded analytic functions on \({\mathbb{D}}\), is bounded or compact. When the symbol is univalent, these become questions in univalent function theory. The corresponding problems for the companion operator, \({ S_g f(z)= \int_0^z f'(t)g(t) \,dt}\), acting on H ∞ are also studied.

Brennan’s conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk \({\mathbb{D}}\) onto a simply connected domain G and −1/3 < p < 1, then 1/(τ′) p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces \({L^2_a(\mu _p)}\) on G and prove that Brennan’s conjecture is equivalent to the existence of compact composition operators on these spaces for every simply connected domain G and all \({p\in(-1/3,1)}\). Motivated by this result, we study the boundedness and compactness of composition operators in this setting.

Article history: Received 16 March 2011 Accepted 19 April 2011 Available online 5 May 2011 Presented by Gilles Pisier An example is constructed of two Riemann maps φ and ψ of the unit disk onto the same domain such that φ′/ψ ′ is bounded but not bounded away from zero. This is shown by producing explicit analytic expressions of φ and ψ . © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é On construit un exemple de deux applications conformes φ et ψ du disque unité sur le même domaine telles que le rapport φ′/ψ ′ soit borné et le rapport ψ ′/φ′ non borné. On donne pour cela des expressions analytiques explicites pour φ, ψ . © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. Version française abrégée Le but de cette Note est de construire un exemple d’application conforme lié à la Conjecture de Brennan [2]. La Conjecture de Brennan est une conjecture importante dans la théorie des fonctions univalentes qui concerne le degré d’intégrabilité de la dérivée d’une application conforme g d’un domaine simplement connexe planaire G sur le disque unité D de C. Cette conjecture dit que ∫ G ∣∣g′∣∣t dA < ∞ (1) pour 4/3 < t < 4. Il y a des classes d’applications conformes pour lesquelles la conjecture est démontrée, voir [1] par exemple. Un progrès important a été réalisé dans [4]. On trouvera dans [6] une formulation équivalente de la Conjecture de Brennan en termes d’opérateurs de composition agissant sur certains espaces de Hilbert de fonctions analytiques sur G . Posons dμp = |g′|2p+2 dA pour un réel fixé p, et soit L2 a(μp) := { F ∈ H(G): ‖F‖2 = ∫

A holomorphic self-map φ \varphi of the unit disk is constructed such that the composition operator induced by φ \varphi is bounded on the Hardy-Sobolev space H 2 1 H^1_2 of order 2 2 as well as on the ordinary holomorphic Lipschitz space Lip 1 \textrm {Lip}_1 but unbounded on the Zygmund class Λ 1 \Lambda _1 . Among these three function spaces we have embedding relations H 2 1 ⊂ Lip 1 ⊂ Λ 1 H^1_2\subset \textrm {Lip}_1\subset \Lambda _1 . So, the main points here are that our construction provides a composition operator which is bounded on smaller spaces, but not on a larger space and that all the function spaces involved are standard ones.

An example is constructed of two Riemann maps phi and psi of the unit disk onto the same domain such that phi'/psi' is bounded but not bounded away from zero. This is shown by producing explicit analytic expressions of phi and psi. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Let 1 ≤ p < ∞ and let μ be a finite positive Borel measure on the unit disk D . The area Nevanlinna-Lebesgue space N p ( μ ) consists of all measurable functions h on D such that log + | h | ∈ L p ( μ ), and the area Nevanlinna space N α p is the subspace consisting of all holomorphic functions, in N p ((1−| z | 2 ) α dv ( z )), where α > −1 and ν is area measure on D . We characterize Carleson measures for N α p , defined to be those measures μ for which N α p ⊂ N p (μ). As an application, we show that the spaces N α p are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given.

We study the composition operator CΨ induced by an analytic self-map Ψ of the unit ball BN in CN that extends to be smooth on B¯N. When Ψ is of class C3 on B¯N, we extend to weighted Bergman spaces W.R. Wogen's characterization of when CΨ is bounded on Hp(BN). Next, when Ψ is of class C4 on B¯N, we show that if ϵ>0 and CΨ:Aαp(BN)→Aα+1/4−ϵp(BN), then CΨ is bounded on Aαp(BN). The discrete jump of size 1/4 in the exponent of the weight is sharp. Examples are given that show the assumption Ψ is smooth is essential in these theorems.

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.