Let $φ$ be a holomorphic self-map of a bounded homogeneous domain $D$ in $\mathbb{C}^n$. In this work, we show that the composition operator $C_φ: f\mapsto f\circ φ$ is bounded on the Bloch space $\mathcal{B}$ of the domain and provide estimates on its operator norm. We also give a sufficient condition for $φ$ to induce an isometry on $\cal{B}$. This condition allows us to construct non-trivial examples of isometric composition operators in the case when $D$ has the unit disk as a factor. We then obtain some necessary conditions for $C_φ$ to be an isometry on $\cal{B}$ when $D$ is a Cartan classical domain. Finally, we give the complete description of the spectrum of the isometric composition operators in the case of the unit disk and for a wide class of symbols on the polydisk.
In this paper we consider composition operators on Harmonic-Bloch type spaces and we compute the spectrum of composition operators. Also, we characterize isometric composition operators on harmonic Bloch type spaces.
We investigate the isometric composition operators on the analytic Besov spaces. For $1<p<2$ we show that an isometric composition operator is induced only by a rotation of the disk. For $p>2$, we extend previous work on the subject. Finally, we analyze this same problem for the Besov spaces with an equivalent norm.
The aim of this paper is to investigate $m$--isometric composition operators on directed graphs with one circuit. We establish a characterization of $m$--isometries and prove that complete hyperexpansiveness coincides with $2$--isometricity within this class. We discuss the $m$--isometric completion problem for unilateral weighted shifts and for composition operators on directed graphs with one circuit. The paper is concluded with an affirmative solution of the Cauchy dual subnormality problem in the subclass with circuit containing one element.
We show that a composition operator on weighted Bergman spaces $\mathcal{A}_μ^p$ is invertible if and only if it is Fredholm if and only if it is an isometry.
In this paper we characterize essential norm of composition operators on the spaces of Harmonic Bloch functions. These results extends the similar results that were proven for composition operators on Bloch spaces.
Given two metric spaces $M$ and $N$ we study, motivated by a question of N. Weaver, conditions under which an isometric composition operator $C_ϕ:\mathrm{Lip}_0(M)\longrightarrow \mathrm{Lip}_0(N)$ is isometric depending on the properties of $ϕ$. We obtain a complete characterisation of those operators $C_ϕ$ in terms of a property of the function $ϕ$ in the case that $B_{\mathcal F(M)}$ is the closed convex hull of its preserved extreme points. Also, we obtain necessary and sufficient conditions for $C_ϕ$ being isometric in the case that $M$ is geodesic.
In this paper we characterize some basic properties of composition operators on the spaces of harmonic Bloch functions. First we provide some equivalent conditions for boundedness and compactness of composition operators. Then by using these conditions we estimate the essential norm of composition operators. These results extends the similar results that were proven in the literature on the Bloch spaces.
The spectra of invertible weighted composition operators $uC_φ$ on the Bloch and Dirichlet spaces are studied. In the Bloch case we obtain a complete description of the spectrum when $φ$ is a parabolic or elliptic automorphism of the unit disc. In the case of a hyperbolic automorphism $φ$, exact expressions for the spectral radii of invertible weighted composition operators acting on the Bloch and Dirichlet spaces are derived.
Let $S(\mathbb{D})$ be the collection of all holomorphic self-maps on $\mathbb{D}$ of the complex plane $\mathbb{C}$, and $C_φ$ the composition operator induced by $φ\in S(\mathbb{D})$. We obtain that there are no hypercyclic composition operators on the little Bloch space $\mathcal{B}_0$ and the Besov space $B_p$.
The purpose of this paper is to describe the characterization for the compact difference of two composition operators acting between analytic Besov spaces and the weighted little Bloch type space over the unit disk.
Given pointed metric spaces $X$ and $Y$, we characterize the basepoint-preserving Lipschitz maps $ϕ$ from $Y$ to $X$ inducing an isometric composition operator $C_ϕ$ between the Lipschitz spaces $Lip_0(X)$ and $Lip_0(Y)$, whenever $X$ enjoys the peak property. This gives an answer to a question posed by N. Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].
Cauchy dual Subnormality of extensions of $m$-isometric composition operators on directed graphs
Sree Narayana College Alathur, Affiliated to University of Calicut（阿拉图尔斯里纳拉亚纳学院（隶属于卡利卡特大学））；Nalanda University（那烂陀大学）；University College Thiruvananthapuram（蒂鲁瓦南特普拉姆大学学院）
探讨有一个回路和多个分支顶点的有向图上\(k -\)拟\( - m -\)等距复合算子柯西对偶的解析性、\(\Delta_{C_\phi}\)-正则性、核条件及次正规性。
Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.
We determine both the semigroup and spectral properties of a group of weighted composition operators on the Little Bloch space. It turns out that these are strongly continuous groups of invertible isometries on the Bloch space. We then obtain the norm and spectra of the infinitesimal generator as well as the resulting resolvents which are given as integral operators. As consequences, we complete the analysis of the adjoint composition group on the predual on the nonreflexive Bergman space, and a group of isometries associated with a specific automorphism of the upper half plane.
We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide computable estimates on the operator norm.
In this article, we estimate the essential norm of weighted composition operator $W_{u, φ}$, acting on the logarithmic Bloch space $\mathcal{B}^{v_{\log}}$, in terms of the $n$-power of the analytic function $φ$ and the norm of the $n$-power of the identity function. Also, we estimate the essential norm of the weighted composition operator from $\mathcal{B}^{v_{\log}}$ into the growth space $H_{v_{\log}}^{\infty}$. As a consequence of our result, we estimate the essential norm of the composition operator $C_φ$ acting on the Logarithmic-Zygmund space.
In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.
In this paper, we completely characterize nuclear Volterra composition operators $T^ϕ_g : \mathcal H^\infty_ν\longrightarrow \mathcal H^\infty_μ$ and $S^ϕ_g : \mathcal H^\infty_ν\longrightarrow \mathcal H^\infty_μ$ acting between weighted type spaces in terms of the symbols $g$ and $ϕ$ of $T^ϕ_g$ and $S^ϕ_g$ and weights $ν$ and $μ$, when the weights $ν$ and $μ$ are normal weights in the sense of Shields and Williams. Moreover, nuclear Volterra composition operators acting between little weighted type spaces and Bloch spaces of order $β$ are also characterized.
Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$. In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space $H^\infty(D)$ into the Bloch space of $D$. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if $D$ is a bounded symmetric domain, the bounded multiplication operators from $H^\infty(D)$ to the Bloch space of $D$ are the operators wh…
In this paper, we study the multiplication operators on the Bloch space of a bounded homogeneous domain in $\mathbb{C}^n$. Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. We prove that the only bounded multiplication operators on the Bloch space of the polydisk are those whose symbol is constant. Furthermore, we prove that for a large class of bounded symmetric domains, the isometric multiplication operators are those whose symbol is a constant of modulus one.
Let $u$ be a holomorphic function and $φ$ a holomorphic self-map of the open unit disk $\mathbb{D}$ in the complex plane. We give some new characterizations for the boundedness of the weighted composition operators $uC_φ$ from Bloch type spaces to Zygmund type spaces in $\mathbb{D}$ in terms of $u, φ$, their derivatives and the $n$-th power $φ^n$ of $φ$. Moreover, we obtain some similar estimates for their essential norms. From which the sufficient and necessary conditions of compactness of the operators $uC_φ$ follows immediately.
We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like $BMOA$, $Q_α$, and analytic Besov spaces $B^p$. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding "small" spaces we also characterize the boundedness and show that it is equivale…
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.
We consider abstract Banach spaces of analytic functions on general bounded domains that satisfy only a minimum number of axioms. We describe all invertible (equivalently, surjective) weighted composition operators acting on such spaces. For the spaces of analytic functions in the disk whose norm is given in terms of a natural seminorm (such as the typical spaces given in terms of derivatives), we describe all surjective weighted composition operators that are isometric. This generalizes a number of known results.
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 $H^\infty$.
The properties of Volterra-composition operators on the weighted Bergman space with exponential type weights are investigated in this paper. We state some necessary and sufficient conditions that a Volterra-composition operator from the weighted Bergman space to Bloch type space ( or little Bloch type space ) must satisfy for the Volterra-composition operator to be bounded or compact.
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.
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…
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 $β^α.$
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 study the weighted composition operators between the Lipschitz space and the space of bounded functions on the set of vertices of an infinite tree. We characterized the boundedness, the compactness, and the boundedness from below of weighted composition operators. We also determine the isometric weighted composition operators.
We study composition operators on the weighted Banach spaces of an infinite tree. We characterize the bounded and the compact operators, as well as determine the operator norm and the essential norm. In addition, we study the isometric composition operators.
We obtain necessary and sufficient conditions for the composition and weighted composition operator and product of composition operators to be isometry and unitary on $H_{E}(ξ).$ With the help of counter example we also prove that the product of two non isometric composition operator and weighted composition operator can be isometry on $H_{E}(ξ)$. We also completely characterize the boundedness of generalized weighted composition operators on $H_{E}(ξ).$
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 study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. We study the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
We study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. The central issue is to understand the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
We identify the predual of the nonreflexive Bergman space of the upper half plane, $L_a^1(\uP,μ_{\al})$, with the little Bloch space of the upper half plane consisting of functions vanishing at $i$. We then investigate both the semigroup and spectral properties of the adjoint groups of composition operators which are naturally obtained from the duality pairing and are therefore defined on the identified predual.
We consider weighted composition operators on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of weighted composition operators on general function spaces, and in particular on weak vector-valued spaces. As an application, we characterize the weak compactness of these operators between two different vector-valued Bloch-type spaces. This result appears to be new also in the scalar-valued case.
Let $Ω_1,Ω_2\subset {\mathbb C}$ be bounded domains. Let $ϕ:Ω_1\rightarrow Ω_2$ holomorphic in $Ω_1$ and belonging to $W^{1,\infty}_{Ω_2}(Ω_1)$. We study the composition operators $f\mapsto f\circϕ$ on generalized Hardy spaces on $Ω_2$, recently considered in \cite{bfl, BLRR}. In particular, we provide necessary and/or sufficient conditions on $ϕ$, depending on the geometry of the domains, ensuring that these operators are bounded, invertible, isometric or compact. Some of our results are new even for Hardy spaces of analytic functions.
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 give a complete characterization of the spectrum of composition operators, induced by an automorphism of the open unit disk, acting on a family of Banach spaces of analytic functions that includes the Bloch space and BMOA. We show that for parabolic and hyperbolic automorphisms, the spectrum is the unit circle. For the case of elliptic automorphisms, the spectrum is either the unit circle or a finite cyclic subgroup of the unit circle.
In this paper, we introduce a discrete analogue of weighted Hardy spaces on rooted trees and study weighted composition operators between them in detail. In particular, we characterize bounded and compact weighted composition operators between discrete Hardy spaces. We also consider isometric weighted composition operators between these spaces.
In this paper, we study the weighted composition operator on the Fock space $\mf$ of slice regular functions. First, we characterize the boundedness and compactness of the weighted composition operator. Subsequently, we describe all the isometric composition operators. Finally, we introduce a kind of (right)-anti-complex-linear weighted composition operator on $\mf$ and obtain some concrete forms such that this (right)-anti-linear weighted composition operator is a (right)-conjugation. Specially, we present equivalent conditions ensuring weighted composition operators which are conjugate $\mat…
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.
The aim of this paper is to study the Wold-type decomposition in the class of $m$-isometries. One of our main results establishes an equivalent condition for an analytic $m$-isometry to admit the Wold-type decomposition for $m\ge2$. In particular, we introduce the $k$-kernel condition which we use to characterize analytic $m$-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for $m\ge2$. As a result, we also show that $m$-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to…
We prove that in a large class of Banach spaces of analytic functions in the unit disc $\mathbb{D}$ an (unbounded) operator $Af=G\cdot f'+g\cdot f$ with $G,\, g$ analytic in $\mathbb{D}$ generates a $C_0$-semigroup of weighted composition operators if and only if it generates a $C_0$-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces $X $ of analytic functions in $\mathbb{D}$…
