Zehua Zhou, Yan Liu
Let $Ω$ be a bounded symmetric domain except the two exceptional domains of ${\Bbb C}^N$ and $ϕ$ a holomorphic self-map of $Ω.$ This paper gives a sufficient and necessary condition for the composition operator $C_ϕ$ induced by $ϕ$ to be compact on the Bloch space $β(Ω)$.
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In this paper, we study the boundedness and compactness of the differences of two weighted composition operators acting from $α$-Bloch space to $β$-Bloch space on the open unit disk. This study has a relationship to the topological structure of weighted composition from $α$-Bloch space to $β$-Bloch space.
In this paper, we study the boundedness and essential norms of the differences of two generalized composition operators acting from $α$-Bloch space to $β$-Bloch space on the open unit disk. From essential norms, we get the compactness of the differences of two generalized composition operators. This study has a relationship to the topological structure of generalized composition operators acting from $α$-Bloch space to $β$-Bloch space.
Let $ϕ$ be an analytic self-map and $u$ be a fixed analytic function on the open unit disk $D$ in the complex plane $\CC.$ The weighted composition operator is defined\break by \begin{equation*} uC_ϕf =u \cdot (f\circ ϕ), f \in H(D). \end{equation*} Weighted composition operators from Bergman-type spaces into Bloch spaces and little Bloch spaces are characterized by function theoretic properties of their inducing maps.
The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.
Suppose $φ$ is a holomorphic self map of the unit disk and $C_φ$ is a composition operator with symbol $φ$ that fixes the origin and $0<|φ'(0)|<1$. This work explores sufficient conditions that ensure all holomorphic solutions of Schröder equation for the composition operator $C_φ$ belong to a Bloch-type space $\mathcal{B}_α$ for some $α>0$. The results from composition operators have been extended to weighted composition operators in the second part of this work.
Let $ϕ(z)=(ϕ_1(z), ...,ϕ_n(z))$ be a holomorphic self-map of $U^n$ and $ψ(z)$ a holomorphic function on $U^n,$ where $U^n$ is the unit polydisk of ${\Bbb C}^n.$ Let $p\geq 0,$ $q\geq 0$, this paper gives some necessary and sufficient conditions for the weighted composition operator $W_{ψ,ϕ}$ induced by $ψ$ and $ϕ$ to be bounded and compact between $p$-Bloch space ${\cal B}^p(U^n)$ and $q$-Bloch space ${\cal B}^q(U^n).$
Hölder continuity and composition operators in pluriharmonic Bloch spaces
该研究针对复空间单位球上的多调和Bloch映射，证明了对应于Bergman度量的Hölder估计，进而得到多调和Bloch空间复合算子下方有界的充分条件及全纯映射Bloch空间复合算子下方有界的必要条件。
We give some new estimates for the norm and essential norm of a weighted composition operator on the Bloch space. As corollaries, we obtain some new characterizations of the boundedness and compactness of a weighted composition operator on the Bloch space.
In this paper, we give two new characterizations for the boundedness and compactness of the difference of two weighted composition operators acting from $H^\infty$ to the Bloch space.
The boundedness and compactness of weighted composition operators from $H^\infty$ to the Bloch space in the unit ball of Cn are investigated in this paper. In particular, some new characterizations for the boundedness and the essential norm of weighted composition operators are given.
In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space.
We found several new equivalent characterizations for the boundedness and compactness of the differences of weighted differentiation composition operators from Bloch-type space to weighted-type space. Especially, we estimated its essential norm in terms of the $n$-th power of the induced analytic self-maps on the unit disk.
In this paper, we present the current results in the study of weighted composition operators on the Bloch space of bounded homogeneous domains in $\mathbb{C}^n$ with particular emphasis on the issues of boundedness and compactness. We also discuss the bounded and the compact weighted composition operators from the Bloch space to the Hardy space $H^\infty$.
On the space of bounded analytic functions and the Bloch space on the unit disk, we study the compact intertwining relations for composition operators, whose intertwining operators are Volterra type operators. Further, we consider the compact intertwining relations, which are between the whole collection of composition operators and some Volterra operator, and the whole collection of bounded Volterra operators and some composition operator.
Characterizations of composition operators on Bloch and Hardy type spaces
本文利用一般倍增权重和新方法刻画不同区域上Bloch型、Hardy型及Lipschitz型空间之间的复合算子，并给出有界性、紧性刻画及对已有结果的推广。
Let p,q>0. We extend to the n-polydisk previous one-variable characterization results of K. Madigan on the $p$-Lipschitz space and K. Madigan/A. Matheson on the Bloch space by obtaining function-theoretic conditions on a holomorphic self-map of the polydisk such that the induced composition operator is bounded or compact between p- and q-Bloch spaces of the polydisk. These conditions turn out to be different in the cases when p is in (0,1) and when p is at least 1. We also obtain corresponding characterization results for composition operators between generalized little p- and q-Bloch spaces o…
Let $ϕ$ and $ψ$ be holomorphic self-maps of the unit polydisc $U^n$ in the $n$-dimensional complex space, and denote by $C_ϕ$ and $C_ψ$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_ϕ-C_ψ$ from Bloch space to bounded holomorphic function space in the unit polydisc. Moreover the compactness of the difference is also characterized.
Let $λ_i (i=1,...,k)$ be any nonzero complex scalars and $φ_i (i=1,..,k)$ be any analytic self-maps of the unit disk $\mathbb{D}$. We show that the operator $\sum_{i=1}^kλ_iC_{φ_i}$ is compact on the Bloch space $\mathcal{B}$ if and only if $$\lim_{n\to\infty}\|λ_1φ_1^n+λ_2φ_2^n+...+λ_kφ_k^n\|_{\mathcal{B}}=0.$$ We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions.
When 0<p<1, it is known that the p-Bloch and (1-p)-Lipschitz spaces of the unit ball in n-dimensional complex Eucllidean space are equal as sets. We prove that these spaces are additionally norm-equivalent, thus extending known results for n=1 and the polydisk. As an application, we generalize work by Madigan on the disk by investigating boundedness of composition operators between p- and q-Lipschitz spaces of the ball.
We characterize the analytic self-maps $ϕ$ of the unit disk ${\Bbb D}$ in ${\Bbb C}$ that induce continuous composition operators $C_ϕ$ from the log-Bloch space $\mathcal{B}^{\log}({\Bbb D})$ to $μ$-Bloch spaces ${\mathcal B}^μ({\Bbb D})$ in terms of the sequence of quotients of the $μ$-Bloch semi-norm of the $n$th power of $ϕ$ and the log-Bloch semi-norm (norm) of the $n$th power $F_n$ of the identity function on ${\Bbb D}$, where $μ:{\Bbb D}\rightarrow (0,\infty)$ is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of $C_ϕ$ between these spaces, t…
Hölder Continuity and Boundedness of Composition Operators on Pluriharmonic $α$-Bloch Spaces
本文针对复单位球上的多调和α-Bloch空间，结合局部实部估计与次调和性建立全局下界，证明不变权重量的Hölder连续性，并给出复合算子有下界的最优充要采样条件。
Let $ϕ(z)=(ϕ_1(z),...,ϕ_n(z))$ be a holomorphic self-map of $B$ and $ψ(z)$ a holomorphic function on $B$, where $B$ is the unit ball of ${\Bbbb C}^n$. Let $0<p,s<+\infty, -n-1<q<+\infty, q+s>-1$ and $α\geq 0,$ this paper gives some necessary and sufficient conditions for the weighted composition operator $\wco$ induced by $ϕ$ and $ψ$ to be bounded and compact between the space $\Fs$ and $α$-{\sl Bloch} space $β^α.$
The boundedness of compactness of integral-type operators from Hardy space to Bloch space on the upper half-plane $Π_+=\{z\in\mathbb{C}:Imz>0\}$ are characterized.
Let $U^{n}$ be the unit polydisc of ${\Bbb C}^{n}$ and $ϕ=(ϕ_1, >..., ϕ_n)$ a holomorphic self-map of $U^{n}.$ By ${\cal B}^p(U^{n})$, ${\cal B}^p_{0}(U^{n})$ and ${\cal B}^p_{0*}(U^{n})$ denote the $p$-Bloch space, Little $p$-Bloch space and Little star $p$-Bloch space in the unit polydisc $U^n$ respectively, where $p, q>0$. This paper gives the estimates of the essential norms of bounded composition operators $C_ϕ$ induced by $ϕ$ between ${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$) and ${\cal B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$). As their applic…
In this note, we study the boundedness and compactness of integral operators $I_g$ and $T_g $ from analytic Morrey spaces to Bloch space. Furthermore, the norm and essential norm of those operators are given.
In this paper, we establish a Bloch-type growth theorem for generalized Bloch-type spaces and discuss relationships between Dirichlet-type spaces and Hardy-type spaces on certain classes of complex-valued functions. Then we present some applications to non-homogeneous Yukawa PDEs. We also consider some properties of the Lipschitz-type spaces on certain classes of complex-valued functions. Finally, we will study a class of composition operators on these spaces.
In this paper, we establish two new versions of Landau-type theorems for pluriharmonic mappings with a bounded distortion. Then using these results, we derive three Bloch-type theorems of pluriharmonic mappings, which improve the corresponding results of Chen and Gauthier.
Any bounded analytic function $g$ induces a bounded integral operator $S_g$ on the Bloch space, the Dirichlet space and $BMOA$ respectively. $S_g$ attains its norm on the Bloch space and $BMOA$ for any $g$, but does not attain its norm on the Dirichlet space for non-constant $g$. Some results are also obtained for $S_g$ on the little Bloch space, and for another integral operator $T_g$ from the Dirichlet space to the Bergman space.
We obtain a necessary and sufficient condition for the operator of integration to be bounded on $H^\infty$ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. This gives a full characterization of symbols of certain Volterra operators that act on bounded analytic functions in the disc if the symbol is assumed to be univalent. Without this assumption the answer is not known, and as the example at the end of the paper shows, the natural answer is definitely false.
Let $μ$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_μ=(μ_{n+k})_{n,k\geq 0}$ with entries $μ_{n,k}=μ_{n+k}$ induces the operator $$ \mathcal{H}_μ(f)(z)=\sum^\infty_{n=0}\left(\sum^\infty_{k=0}μ_{n,k}a_k\right)z^n $$ on the space of all analytic functions $f(z)=\sum^\infty_{n=0}a_nz^n$ in the unit disk $\mathbb{D}$. In this paper, we characterize the boundedness and compactness of $\mathcal{H}_μ$ from Bloch type spaces to the BMOA and the Bloch space. Moreover we obtain the essential norm of $\mathcal{H}_μ$ from $α$ Bloch type spaces to Bloch space and BM…
Let $μ$ be a positive Borel measure on the interval [0,1). For $α>0$, the Hankel matrix $\mathcal{H}_{μ,α}=(μ_{n,k,α})_{n,k\geq 0}$ with entries $μ_{n,k,α}=\int_{[0,1)}\frac{Γ(n+α)}{n!Γ(α)}t^{n+k}dμ(t)$ formally induces the operator $$\mathcal{H}_{μ,α}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n, k,α} a_{k}\right)z^{n} $$ on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ in the unit disc $\mathbb{D}$. In this paper, we characterize the measures $μ$ for which $\mathcal{H}_{μ,α}$ ($α\geq 2$) is a bounded (resp., compact) operator from the Bloch type space $\…
Let $ E $ be a space of holomorphic functions on the unit ball $ B_X $ of a Banach space $ X.$ In this work, we introduce a Banach structure associated to $ E $ on the linear space $ WE(Y) $ containing $ Y$-valued holomorphic functions on $ B_X $ such that $ w \circ f \in E $ for every $ w \in W, $ a separating subspace of the dual $ Y' $ of a Banach $ Y. $ We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators $ W_{ψ,φ}: f\mapsto ψ\cdot(f \circ φ) $ on $ E $ and $ \widetilde{W}_{ψ,φ}: g\mapsto ψ\cdot(g \circ φ) $ on $ WE(Y)$ via some ch…
In this paper we begin the study of some important Banach spaces of slice hyperholomorphic functions, namely the Bloch, Besov and weighted Bergman spaces, and we also consider the Dirichlet space, which is a Hilbert space. The importance of these spaces is well known, and thus their study in the framework of slice hyperholomorphic functions is relevant, especially in view of the fact that this class of functions has recently found several applications in operator theory and in Schur analysis. We also discuss the property of invariance of this function spaces with respect to Moebius maps by usi…
Let $\mathbb{B}$ be the unit ball of a complex Banach space $X$. In this paper, we will generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $\mathbb{B}$ by using the radial derivative. Next, we define an extended Cesàro operator $T_φ$ with holomorphic symbol $φ$ and characterize those $φ$ for which $T_φ$ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those $φ$ for which $T_φ$ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol $φ$. When $\mat…
Let $H(\mathbb{B})$ denote the space of all holomorphic functions on the unit ball $\mathbb{B}\in \mathbb{C}^n$. In this paper we investigate the boundedness and compactness of the products of radial derivative operator and the following integral-type operator $$ I_ϕ^g f(z)=\int_0^1 \Re f(ϕ(tz))g(tz)\frac{dt}{t},\ z\in\mathbb{B} $$ where $g\in H(\mathbb{B}), g(0)=0$, $ϕ$ is a holomorphic self-map of $\mathbb{B}$,\ between Zygmund spaces and Bloch spaces.
When the weight $μ$ is more general than normal, the complete characterizations in terms of the symbol $g$ and weights for the conditions of the boundedness and compactness of $T_g: H^{\infty}_ν\rightarrow H^{\infty}_μ$ and $S_g: H^{\infty}_ν\rightarrow H^{\infty}_μ$ are still unknown. Smith et al. firstly gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, continuing their lines of investigations, we give the complete characterizations of the condition…
Denote by $ B_X $ the unit ball of an infinite-dimensional complex Hilbert space $ X. $ Let $ψ\in H(B_X),$ the space of all holomorphic functions on the unit ball $B_X,$ $φ\in S(B_X)$ the set of holomorphic self-maps of $B_X. $ Let $C_{ψ, φ}: \mathcal B_ν(B_X)$ (and $ \mathcal B_{ν,0}(B_X)$) $\to \mathcal B_μ(B_X) $ (and $ \mathcal B_{μ,0}(B_X)$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $ C_φ$ and integral operator $T_ψ.$ In this paper, we characterize the boundedness and compactness of $ C_{ψ,φ} $ via the estimates for the restrictions of $ ψ$ …
We study the boundedness of composition operators on the bidisk using reproducing kernels. We show that a composition operator is bounded on the Hardy space of the bidisk if some associated function is a positive kernel. This positivity condition naturally leads to the study of the sub-Hardy Hilbert spaces of the bidisk, which are analogs of de Branges-Rovnyak spaces on the unit disk. We discuss multipliers of those spaces and obtain some classes of bounded composition operators on the bidisk.
We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.
In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space $\mathcal{B}(\mathbb{B}^{n})$ and the generalized Bloch space $\mathcal{L}_{\infty,ω}\mathcal{B}_{α,\mathrm{a}}^{0}(\mathbb{B}^{n})$, respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's rel…
Let μbe any weight function defined on the unit disk $\Bbb D$ and let $ϕ$ be an analytic self-map of $\Bbb D$. In the present paper we show that the essential norm of composition operator $C_ϕ$ mapping from the $α$-Bloch space, with $α>0$, to $μ$-Bloch space $\mathcal{B}^μ$ is comparable to $$ \limsup_{|a|\to 1^-}\|σ_a\circϕ\|_{\mathcal{B}^μ}, $$ where, for $a\in\Bbb D$, $σ_a$ is a certain special function in $α$-Bloch space.
We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.
order boundedness and essential norm of generalized weighted composition operators on Bergman spaces with doubling weights
本文研究带倍增权Bergman空间上广义加权复合算子的序有界性和本质范数，并利用降阶方法给出本质范数估计。
We study analogues of well-known relationships between Muckenhoupt weights and $BMO$ in the setting of Bekollé-Bonami weights. For Bekollé-Bonami weights of bounded hyperbolic oscillation, we provide distance formulas of Garnett and Jones-type, in the context of $BMO$ on the unit disc and hyperbolic Lipschitz functions. This leads to a characterization of all weights in this class, for which any power of the weight is a Bekollé-Bonami weight, which in particular reveals an intimate connection between Bekollé-Bonami weights and Bloch functions. On the open problem of characterizing the closure …
A class of symbols that induce bounded composition operators for Dirichlet-type spaces on the disc
本文利用双圆盘加权 Bergman 空间复合算子的多维理论，给出一类单位圆盘全纯自映射符号，证明其在 Dirichlet 型空间上诱导有界复合算子。
Some equivalent definitions for weighted Korenblum and Bloch-type spaces of the upper half-plane
本文通过有界Hankel算子的符号，证明了上半平面加权Korenblum空间和Bloch型空间的等价刻画。
The Bohr Radius of the Weighted Bloch Spaces
本文定义 Banach 空间对的 Bohr 半径，改进 Bloch 空间到有界函数空间的下界，证明任意加权 Bloch 空间的 Bohr 半径不小于 $1/\sqrt{2}$，并给出尖锐性判据及权重示例。
Logarithmic Bloch spaces in the polydisc, endpoint results for Hankel operators and pointwise multipliers
本文研究单位多圆盘中的两类对数 Bloch 空间，利用有界 Hankel 算子的符号给出等价刻画，并完整描述不同 Bloch 空间之间的逐点乘子。
Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces
本文研究非光滑有理内函数诱导的复合算子在两个不同Bergman空间间的有界性，给出了对应参数条件下的一个充分条件。
In this paper, we first study the bounded mean oscillation of planar harmonic mappings, then a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings is established. At last, we obtain sharp estimates on Lipschitz number of planar harmonic mappings in terms of bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to $BMO_{2}$ as a Banach space..
Generalized integral type Hilbert operator acting on weighted Bloch space
本文研究广义积分型Hilbert算子$\mathcal{I}_{\mu_{\alpha+1}}$在加权Bloch空间之间的有界性与紧性，并给出Bloch型空间及对数Bloch空间上的完全刻画。
Restrictions of Békollé--Bonami weights and Bloch functions
研究Békollé-Bonami权重在单位圆子集上的限制，通过Makinov的概率方法将问题转化为dyadicmartingales的限制问题，证明Wolff定理的类比。
Let $g$ be a holomorphic map of $B$, where $B$ is the unit ball of ${C}^n$. Let $0<p<+\infty, -n-1<q<+\infty$, $q>-1$ and $α>0$. This paper gives some necessary and sufficient conditions for the Extended Ces$\acute{a}$ro Operators induced by $g$ to be bounded or compact between generalized Besov space $B(p,q)$ and $α$- Bloch space ${\mathcal B}^α.$
Positive Toeplitz Operators from a Weighted Harmonic Bloch Space into Another
本文在 $\mathbb{R}^n$ 单位球上定义了全参数范围的加权调和 Bloch 空间之间的正 Toeplitz 算子，并利用 Carleson 测度刻画了其有界性与紧致性。
In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.
In this paper, we study weighted composition operators on Bergman spaces of analytic functions which are square integrable on polydisk. We develop the study in full generality, meaning that the corresponding weighted composition operators are not assumed to be bounded. The properties of weighted composition operators such as real symmetry, unitariness, complex symmetry, are characterized fully in simple algebraic terms, involving their symbols. As it turns out, a weighted composition operator having a symmetric structure must be bounded. We also obtain the interesting consequence that real sym…
This paper characterises the boundedness and compactness of Agler--McCarthy monomial operators by reducing them to weighted composition operators and deriving explicit Carleson measure criteria on the half-plane. The results are illustrated by examples.
We completely characterize the mean ergodic composition operators on $H^\infty(\mathbb{B}_n)$. In particular, we show that a composition operator acting on this space is mean ergodic if and only if it is uniformly mean ergodic.
We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore we discuss the relations of such a Bloch type condition with the analogous hyperbolic Lipschitz condition.
Shift invariant subspaces in the Bloch space
本文研究Bloch空间中平移算子的弱星闭不变子空间，证明有界解析函数可分解为循环因子与满足持久性性质的真不变子空间生成因子，奇异内函数起关键作用，并指出与Bergman空间描述的显著差异，同时给出非循环的可逆函数，回答了九十年代初的开放问题。
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