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.
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.
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.
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.
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.
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.
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…
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 $β^α.$
We consider the space $\bk^1_{\log^α}$, of analytic functions on the unit disk $\D,$ defined by the requirement $\int_\D|f'(z)|ϕ(|z|)\,dA(z)<\infty,$ where $ϕ(r)=\log^α(1/(1-r))$ and show that it is a predual of the "$\log^α$-Bloch" space and the dual of the corresponding little Bloch space. We prove that a function $f(z)=\sum_{n=0}^\infty a_nz^n$ with $a_n\downarrow 0$ is in $\bk^1_{\log^α}$ iff $\sum_{n=0}^\infty \log^α(n+2)/(n+1)<\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in $\bk^1_{\log^α}.$ Some properties o…
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…
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…
We consider a class of two-parameter weighted integral operators induced by harmonic Bergman-Besov kernels on the unit ball of $\mathbb{R}^{n}$ and characterize precisely those that are bounded from Lebesgue spaces $L^{p}_α$ into Harmonic Bergman-Besov $b^{q}_β$ or weighted Bloch Spaces $b^{\infty}_β $, for $1\leq p\leq\infty$, $1\leq q< \infty$ and $α,β\in \mathbb{R}$. These operators can be viewed as generalizations of the harmonic Bergman-Besov projections. Also, our results remove the disturbing conditions $β>-1$ when $q<\infty$ and $β\geq 0$ when $q=\infty$ of Doğan (A Class of Integral O…
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 $Ω$ 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 $β(Ω)$.
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 $ϕ(z)=(ϕ_1(z), ...,ϕ_n(z))$ 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 ${\cal B}^q(U^n).$
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 ${\mathcal B}^α.$
Hölder continuity and composition operators in pluriharmonic Bloch spaces
Huaqiao University（华侨大学）；Guangdong Technion - Israel Institute of Technology（广东以色列理工学院）；Shantou University（汕头大学）；Technion - Israel Institute of Technology（以色列理工学院）
When we deal with $H^{\infty}$, it is known that $c_0-$interpolating sequences are interpolating and it is sufficient to interpolate idempotents of $\ell_\infty$ in order to interpolate the whole $\ell_\infty$. We will extend these results to the frame of interpolating sequences for Bloch type spaces $\mathcal{B}_{v}^\infty$ and study the connection between the interpolating operators on $\mathcal{B}_{v}^\infty$ and $\mathcal{B}_v^0$. Furthermore, for some particular weights $v$, we will provide examples of interpolating sequences for $\mathcal{B}_{v}^\infty$ whose constant of separation is as…
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…
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…
We consider weighted Bergman projection $P_α: L^{\infty}(\Bbb B) \rightarrow {\cal B} $ where $α>-1$ and $\cal B$ is the Bloch space of the unit ball $\Bbb B$ of the complex space $\Bbb C^n.$ We obtain the exact norm of the operator $P_α$ where the Bloch space is observed as a space with norm (and semi-norm) induced from the Besov space $B_{p},0<p<\infty,(B_{\infty}=\cal B).$ Our work contains, as a special case, the main results from \cite{Kalaj} and \cite{Ant}.
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$ …
For $0<p<\infty $ we let $\mathcal D^p_{p-1}$ denote the space of those functions $f$ which are analytic in the unit disc $\mathbb D $ and satisfy $\int_\mathbb D (1-| z|)\sp {p-1}| f'(z)| \sp p\,dA(z)<\infty $. It is known that, whenever $p\neq q$, the only multiplier from $\mathcal D^p_{p-1} $ to $\mathcal D^q_{q-1} $ is the trivial one. However, if $X$ is a subspace of the Bloch space and $0<p\le q<\infty$, then $X \cap \mathcal D^p_{p-1}\subset X\cap \mathcal D^q_{q-1} $, a fact which implies that the space of multipliers $\M(\mathcal D^p_{p-1}\cap X, \mathcal D^q_{q-1} \cap X)$ is non-tri…
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.
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 $\…
We give new constructions of pair of functions $(f, g)$, analytic in the unit disc, with $g\in H^\infty $ and $f$ an unbounded Bloch function, such that the product $g\cdot f$ is not a Bloch function.
In this paper, we study (uniformly) mean ergodic composition operators on $H^\infty(\mathbb{B}_n)$. Under some additional assumptions, it is shown that mean ergodic operators have norm convergent iterates in $H^\infty(\mathbb{B}_n)$, and that they are always uniformly mean ergodic.
Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v $ that fulfill a doubling condition. In the two-dimensional case let $u (re^{i\ta},ξ) = \sum_{j=0}^\infty (a_{j0} ξ_{j0} r^j \cos jθ+a_{j1} ξ_{j1} r^j \sin jθ)$ where $ξ=\{ξ_{ji}\}%_{k=0}^\infty $ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji} $ wh…
In this article, we introduce the class $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ of harmonic $α$-Bloch-type mappings on $Ω$ as a generalization of the class $\mathcal{B}_{\mathcal{H},Ω}(α)$ of harmonic $α$-Bloch mappings on $Ω$, where $Ω$ is arbitrary proper simply connected domain in the complex plane. We study several interesting properties of the classes $\mathcal{B}_{\mathcal{H},Ω}(α)$ and $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ on arbitrary proper simply connected domain $Ω$ and on the shifted disk $Ω_γ$ containing $\mathbb{D}$, where $$ Ω_γ:=\bigg\{z\in\mathbb{C} : \bigg|z+\fracγ{1-γ}\bigg|<\fr…
A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.
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.
For analytic functions $g$ on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ from a space $X$ of analytic functions in the unit disc to $H^\infty$, in terms of neat and useful conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA spaces.
We obtain some estimates for norm and essential norm of the difference of two composition operators between weighted Bergman spaces $A^p_α$ and $A^q_β$ on the unit ball. In particular, we completely characterize the boundedness and compactness of $C_φ-C_ψ: A^p_α\to A^q_β$ for full range $0<p,q<\infty$, $-1<α,β<\infty$.
We study the holomorphic tent spaces $\mathcal{HT}^p_{q,α}(\Bn)$, which are motivated by the area function description of the Hardy spaces on one hand, and the maximal function description of the Hardy spaces on the other. Characterizations for these spaces under general fractional differential operators are given. We describe the dual of $\mathcal{HT}^p_{q,α}(\Bn)$ for the full range $0<p,q<\infty$ and $α>-n-1$. Here the case $1<p<\infty$, $q\leq 1$ leads us to the Hardy-Bloch type spaces $\mathcal{BT}^{p}(\Bn)$ and $\mathcal{BT}^{p,0}(\Bn)$, the latter being the predual of $\mathcal{HT}^p_{1…
Let $f$ be a complex-valued harmonic mapping defined in the unit disk $\mathbb D$. We introduce the following notion: we say that $f$ is a Bloch-type function if its Jacobian satisfies $$ \sup_{z\in\mathbb D}(1-|z|^2)\sqrt{|J_f(z)|}<\infty. $$ This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which states that an analytic $φ$ is Bloch if and only if there exists $c>0$ and a univalent $ψ$ such…
We characterize the semigroups of composition operators that are strongly continuous on the mixed norm spaces $H(p,q,α)$. First, we study the separable spaces $H(p,q,α)$ with $q<\infty,$ that behave as the Hardy and Bergman spaces. In the second part we deal with the spaces $H(p,\infty,α),$ where polynomials are not dense. To undertake this study, we introduce the integral operators, characterize its boundedness and compactness, and use its properties to find the maximal closed linear subspace of $H(p,\infty,α)$ in which the semigroups are strongly continuous. In particular, we obtain that thi…
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…
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…
This paper deals with the the norm of the weighted Bergman projection operator $P_α:L^\infty\to \mathcal{B}$ where $α>-1$ and $\mathcal{B}$ is the Bloch space of the unit ball of the complex space $\mathbf{C}^n$. We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper \cite{Perala}.
