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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.

The disjoint properties of finitely many composition operators acting on the weighted Banach spaces of holomorphic functions in the unit disk were investigated in this paper.

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.

We characterize the hypercyclicity of weighted composition operators on the weighted little Bloch space B 0 α , α > 0 .

In this paper, we study the atomic decomposition for variable exponent Bergman spaces. We also give some characterizations for the boundedness and compactness of composition operators on these spaces, which corrects a result by Morovatpoor and Abkar in (Mediterr. J. Math. 17:9, 2020). Furthermore, we investigate the difference of two composition operators on variable exponent Bergman spaces.

This paper characterizes the boundedness and compactness of the differences of weighted differentiation composition operators acting from the ?-Bloch space B? to the space H? of bounded holomorphic functions on the unit disk D.

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 characterize the boundedness and compactness of the differences of weighted differentiation composition operators $$ {D}_{\varphi_1,{u}_1}^n-{D}_{\varphi_2,{u}_2}^n, $$ where n ∈ ℕ0, u1,u2 ∈ H( $$ \mathbb{D} $$ ), and $$ \varphi $$ 1, $$ \varphi $$ 2 2 S( $$ \mathbb{D} $$ ), from mixed-norm spaces H(p, q, ϕ), where 0 < p, q < ∞ and ϕ is normal, to weighted-type spaces H ∞ .

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

The authors characterize the order boundedness of weighted composition operators acting between Dirichlet type spaces.

We discuss the frequent hypercyclicity and hypercyclicity of multiples of the linear fractional composition operators on A α p and give equivalent conditions of the frequent hypercyclicity and hypercyclicity. We also characterize disjoint hypercyclicity and disjoint mixing behavior of finitely many linear fractional composition operators on the weighted Bergman spaces A α p .

In this paper, we investigate when weighted composition operators acting on Dirichlet spaces $\mathcal{D}(\mathbb{B}_{N})$ are complex symmetric with respect to some special conjugations, and provide some characterizations of Hermitian weighted composition operators on $\mathcal{D}(\mathbb{B}_{N})$. Furthermore, we give a sufficient and necessary condition for $J$-symmetric weighted composition operators on Hardy spaces $H^2(\mathbb{B}_{N})$ to be unitary or Hermitian, then some new examples of complex symmetric weighted composition operators on $H^2(\mathbb{B}_{N})$ are obtained. We also discuss the normality of complex symmetric weighted composition operators on $H^2(\mathbb{B}_{N})$., 18 pages

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.

We characterize the difference of derivative weighted composition operators on the Bergman space in the unit disk and determine when linear-fractional derivative weighted composition operators belong to the same component of the space of derivative weighted composition operators on the Bergman space under the operator norm topology.

We give some new characterizations for compactness of weighted composition operators uCφ acting on Bloch-type spaces in terms of the power of the components of φ where φ is a holomorphic self-map of the polydisk 𝔻n, thus generalizing the results obtained by Hyvärinen and Lindström in 2012.

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 characterize disjoint mixing and disjoint hypercyclicity of finite many composition operators acting on the Hardy space on the unit ball.

In this paper, we give some new and simple characterizations for the boundedness and compactness of the integral-type composition operators acting on Bloch-type spaces, and obtain corresponding estimates for their essential norms in terms of the sequence . From which the sufficient and necessary conditions of compactness of the integral-type composition operators follows immediately.

In 2010, Ruhan Zhao obtained the essential norms of composition operators between Bloch type spaces in the disk by the nth power of the induced analytic function. This paper will generalize Zhao’s results to the polydisk. Unlike the case of the composition operators on the unit disk, the essential norms are different for the cases \({p \in (0,1)}\) and p ≥ 1.

In this paper, we characterize the boundedness and compactness of the weighted composition operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.

We investigate the adjoints of linear fractional composition operators C ϕ acting on classical Dirichlet space D(B N ) in the unit ball B N of ℂ N , and characterize the normality and essential normality of C ϕ on D(B N ) and the Dirichlet space modulo constant function D 0(B N ), where ϕ is a linear fractional map ϕ of B N . In addition, we also show that for any non-elliptic linear fractional map ϕ of B N , the composition maps σ o ϕ and ϕ o σ are elliptic or parabolic linear fractional maps of B N .

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.

We study the topological structure of the space of all bounded composition operators from $F(p,q,s)$ to $\mathcal{B}_\mu$ on the unit disk $\mathbb{D}$ in the operator norm topology. At the same time, we characterizes the boundedness and compactness of the differences of two composition operators.

We characterize the isometries on the products of composition and integral operators on the Bloch type space in the disk.

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).