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The spectrum of a weighted composition operator $$C_{\psi , \varphi }$$ that is induced by an automorphism has been investigated for over 50 years. However, many results are got only under the condition that the weight function $$\psi $$ is continuous up to the boundary. In this paper we study the spectra and essential spectra of $$C_{\psi ,\varphi }$$ on weighted Bergman spaces when $$\varphi $$ is an elliptic automorphism, without the assumption that $$\psi $$ is continuous up to the boundary.

This paper identifies a class of complex symmetric weighted composition operators on H2(D) that includes both the unitary and the Hermitian weighted composition operators, as well as a class of normal weighted composition operators identified by Bourdon and Narayan. A characterization of algebraic weighted composition operators with degree no more than two is provided to illustrate that the weight function of a complex symmetric weighted composition operator is not necessarily linear fractional.

Let $$S^p$$ be the space of holomorphic functions whose derivative lies in the classical Hardy space $$H^p$$ over the unit disk. We prove in this paper that the composition operator $$C_\varphi $$ with $$\varphi $$ an automorphism is hypercyclic on $$S^p$$ , $$0

In this article, we provide a complete description of the spectra of linear fractional composition operators acting on the growth space and Bloch space over the upper half-plane. In addition, we also prove that the norm, essential norm, spectral radius and essential spectral radius of a composition operator acting on the growth space are all equal.

Let S ( D ) be the collection of all holomorphic self-maps on D of the complex plane C , and C φ the composition operator induced by φ ∈ S ( D ) . We obtain that there are no hypercyclic composition operators on the little Bloch space B 0 and the Besov space B p .

We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space $L_{a}^{2}(\mathbb{D})$ and weighted Bergman spaces $L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$ when $\unicode[STIX]{x1D6FC}$ is a nonnegative integer.

In this article, we investigate the mean ergodicity of composition operators acting on the Hardy space $$H^p({\mathbb {D}})$$ , $$1\le p

In this paper, we give a new characterization for the boundedness of weighted differentiation composition operator from logarithmic Bloch spaces to Bloch-type spaces and calculate its essential norm in terms of the n-th power of induced analytic self-map on the unit disk. From which a sufficient and necessary condition of compactness of this operator follows immediately.

We investigate isometric composition operators on the weighted Dirichlet space \({D_\alpha }\) with standard weights \({(1 - {\left| z \right|^2})^\alpha },\alpha > - 1\). The main technique used comes from Martin and Vukotic who completely characterized the isometric composition operators on the classical Dirichlet space D. We solve some of these but not in general. We also investigate the situation when \({D_\alpha }\) is equipped with another equivalent norm.

Let u be a holomorphic function and φ a holomorphic self-map of the open unit disk $$\mathbb{D}$$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators uC φ from Zygmund type spaces to Bloch type spaces in $$\mathbb{D}$$ in terms of u, φ, their derivatives, and φ n , the n-th power of φ. Moreover, we obtain some similar estimates for the essential norms of the operators uC φ , from which sufficient and necessary conditions of compactness of uC φ follows immediately.

Let be the class of all holomorphic functions on the unit ball in . For with and an analytic self-map of , the generalized composition operator is defined byIn this paper, we give estimates for the essential norms of the generalized composition operators between Bloch-type spaces in terms of the powers of the components of . We also give new characterizations for the boundedness and compactness of between Bloch-type spaces.

We study the compact intertwining relations for compositionoperators, whose intertwining operators are Volterra type operators fromthe weighted Bergman spaces to the weighted Bloch spaces in the unitdisk. As consequences, we find a new connection between the weightedBergman spaces and little weighted Bloch spaces through this relations. 1. IntroductionIf X and Y are two Banach spaces, the symbol B(X,Y ) denotes the col-lection of all bounded linear operators from X to Y . Let K(X,Y ) be thecollection of all compact elements of B(X,Y ), and Q(X,Y) be the quotientset B(X,Y )/K(X,Y ).For linear operators A ∈ B(X,X), B ∈ B(Y,Y ) and T ∈ B(X,Y ), thephrase “T intertwines A and B in Q(X,Y)” (or “T intertwines A and B com-pactly”) means that(1.1) TA = BT mod K(X,Y ) with T 6= 0 .Notation A ∝ K B (T) represents the relation in equation (1.1). In fact, if T isan invertible operator on X, then the relation ∝ K is symmetric.We denote the class of all holomorphic functions on the complex unit disk Dby H(D), and the collection of all the holomorphic self mappings of Dby S(D).Let α > −1, 0 0. These settingsof α,β,γ and p are valid in the following context unless specifications.

In this article, we characterize the differences of the products of integral type and composition operators acting from the space of bounded functions to the Bloch space in the disc. Corresponding necessary and sufficient conditions are also given for the differences to be bounded and compact.

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

Let H(B) be the space of all holomorphic functions on the unit ball B in CN , and S(B) the collection of all holomorphic self-maps of B . Let φ ∈ S(B) and g ∈ H(B) with g(0) = 0 , the generalized composition operator is defined by C φ ( f )(z) = ∫ 1 0 R f (φ(tz))g(tz) dt t , Here, we characterize the boundedness and compactness of the generalized composition operator acting from Bloch-type spaces Bω and Bω,0 to mixed-norm space H(p,q,φ) on the unit ball B .

This paper discusses the hypercyclicity of weighted composi- tion operators acting on the space of holomorphic functions on the open unit ball BN of CN. Several analytic properties of linear fractional self- maps of BN are given. According to these properties, a few necessary conditions for a weighted composition operator to be hypercyclic in the space of holomorphic functions are proved. Besides, the hypercyclicity of adjoint of weighted composition operators are studied in this paper.

Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C ϕ , 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 spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.

Let φ and ψ be holomorphic self-maps of the unit polydisc Un in the n-dimensional complex space, and denote by Cφ and Cψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators Cφ−Cψ from Bloch space to bounded holomorphic function space in the unit polydisc. The compactness of the difference is also characterized.

Let ϕ(z) = (ϕ1(z),…,ϕn(z)) be a holomorphic self-map of B and ψ(z) a holomorphic function on B, where B is the unit ball of ℂn. Let 0 < p, s < +∞, -n - 1 < q < +∞, q+s > -1 and α ≥ 0, this paper characterizes boundedness and compactness of weighted composition operator Wψ,ϕ induced by ϕ and ψ between the space F(p, q, s) and α-Bloch space [Formula: see text].

Let Un be the unit polydisc of ℂn and φ = ( φ 1 , * , φ n ) a holomorphic self-map of Un. This paper shows that the composition operator C φ induced by φ is bounded on the little Bloch space β 0 * ( U n ) if and only if φ l ∈ β 0 * ( U n ) for every l = 1 , 2 , * , n , and also gives a sufficient and necessary condition for the composition operator C φ to be compact on the little Bloch space β 0 * ( U n ) .

Let Bn be the unit ball of C n and o = o1 … o n a holomorphic self-map of B n . The main purpose of this paper is to discuss the boundedness and compactness of composition operator Co between p-Bloch space βp Bn and q-Bloch space βqBn … wherep p ⩾0 and q⩾0.

Let $S(\mathbb{D})$ be the collection of all the holomorphic self-maps of open unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$, and $C_{\varphi}$, the composition operator induced by $\varphi\in S(\mathbb{D})$. For $\alpha>0,\;\lambda\in \mathbb{C}$, we give some sufficient and necessary conditions for the hypercyclicity of multiples of composition operators $\lambda C_\varphi$ acting on the weighted Banach spaces of entire functions $H_{\alpha,0}^\infty$. Moreover, we obtain a partial characterization for the frequent hypercyclicity of $\lambda C_\varphi$ on $H_{\alpha,0}^\infty$.

Let be the unit ball in the -dimensional complex space, for , a holomorphic function in , and , a holomorphic map from into itself, the weighted composition operator on the weighted Hardy space is given by , where . This paper discusses the spectrum of when it is compact on a certain class of weighted Hardy spaces and when the composition map has only one fixed point inside the unit ball.

Let Un be the unit polydisc of Cn and φ = (φ1,...,φn) a holomorphic self-map of Un. Let 0 ≤α 1. This paper shows that the composition operator C is bounded on the Lipschitz space Lip(Un) if and only if there exists M > 0 such that $$\sum\limits_{k,l = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial zk}}(z)} \right|\left( {\frac{{1 - \left| {z_k } \right|^2 }}{{1 - \left| {\phi _l (z)} \right|^2 }}} \right)^{1 - \alpha } } \leqslant M$$ for z ∈ Un. Moreover Cφ is compact on Lipα(Un) if and only if Cφ is bounded on Lipα(Un) and for every e>0, there exists a δ > 0 such that $$\sum\limits_{k,l = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial zk}}(z)} \right|\left( {\frac{{1 - \left| {z_k } \right|^2 }}{{1 - \left| {\phi _l (z)} \right|^2 }}} \right)^{1 - \alpha } } \leqslant \varepsilon $$ whenever dist((z),∂Un).

We give a new characterization for the boundedness of composition operator followed by differentiation operator acting on Bloch-type spaces and calculate its essential norm in terms of the $n$-th power of the induced analytic self-map on the unit disk. From which some sufficient and necessary conditions of compactness of the operator follow immediately.

On the Bergman space in the unit disk, we study the inter- twining relation for Volterra type operators, whose intertwining operator is a composition operator. We also investigate the \compact" intertwin- ing relations for Volterra type operators. As obvious consequences, the essential commutativity of Volterra type and composition operators are characterized. At the end of the paper, we nd a new connection be- tween the Bergman space and little Bloch space through this essential commutativity.

Let be the unit ball of and a holomorphic self-map of Let , , , , and let be the composition operator between the space and -Bloch space induced by . This paper gives an estimate of the essential norm of . As a consequence, a necessary and sufficient condition for the composition operator to be compact from to is obtained.

In this article, we characterize the surjective isometric composition operator C φ on the weighted Dirichlet-type spaces of the unit disk D , where φ is an analytic self-map of D , and show that C φ is a surjective isometry if and only if φ is a rotation map. 2000 MSC: Primary: 47B33; Secondary: 30D45; 30H30; 30H05; 47B38.