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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.

In this paper, we completely characterize the finite rank commutator and semicommutator of two monomial-type Toeplitz operators on the weighted Bergman space or on the weighted pluriharmonic Bergman space of the unit polydisk. As a consequence, some interesting new phenomenons appear in operator theory on the unit polydisk which are different from that on the unit disk and the unit ball.

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 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.

In the present paper, we investigate the disjoint mixing property of finitely many linear fractional composition operators acting on the space of holomorphic functions on the unit ball in C N , and generalize parts of the results obtained by Bes, Martin and Peris in 2011.

In this paper, analogues of the Berkson–Porta formula for the infinitesimal generators of one-parameter semigroup of holomorphic self-maps on the polydisk are obtained. We give a necessary and sufficient condition for a holomorphic vector field to be an infinitesimal generator which improves the theorem given by Contreras, de Fabritiis and Diaz-Madrigal.

A bounded linear operator $T$ on a Banach space $X$ is called subspace-hypercyclic for a nonzero subspace $M$ if $orb\left( {T,x} \right) \cap M$ is dense in $M$ for a vector $x \in X$, where $orb (T,x)=\{T^nx: n=0,1,2,\cdots\}$. Similarly, the bounded linear operator $T$ on a Banach space $X$ is called subspace-supercyclic for a nonzero subspace $M$ if there exists a vector whose projective orbit intersects the subspace $M$ in a relatively dense set. In this paper we provide a Subspace-Supercyclicity Criterion and offer two equivalent conditions of this criterion. At the same time, we also characterize other properties of subspace-supercyclic operators.

In this paper we give a Carleson measure characterization for the compact composition operators between Dirichlet type spaces. We use this characterization to show that every compact composition operator on Dirichlet type spaces is compact on the Bloch space.

In this paper, we completely determine the commutativity of two Toeplitz operators on the harmonic Bergman space with integrable quasihomogeneous symbols, one of which is of the form $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}e^{ik\theta }r^{\, {m}}$. As an application, the problem of when their product is again a Toeplitz operator is solved. In particular, Toeplitz operators with bounded symbols on the harmonic Bergman space commute with $T_{e^{ik\theta }r^{\, {m}}}$ only in trivial cases, which appears quite different from results on analytic Bergman space in Čučković and Rao [‘Mellin transform, monomial symbols, and commuting Toeplitz operators’, J. Funct. Anal.154 (1998), 195–214].

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.

We discuss the boundedness and compactness of some integral-type operators acting from spaces to mixed-norm spaces on the unit ball of .

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.

In this paper, we study the boundness and compactness of differences of weighted composition operators from the Bloch space B to the space H ∞ of bounded analytic functions on the open unit ball of C n . Moreover, we estimate the norm and the essential norm of differences of composition operators from B to H ∞ .

In this paper, we give a new characterization for the boundedness of the product of differentiation and composition operators \({C_\varphi D^m}\) acting on Bloch-type spaces and obtain an estimate for its essential norm in terms of the sequence \({\{z^n\}^{\infty}_{n=1}}\), from which the sufficient and necessary condition of compactness of the operator \({C_\varphi D^m}\) follows immediately.

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.

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.

In this paper, we limit our analysis to the difference of the weighted composition operators acting from the Hardy space to weighted-type space in the unit ball of ℂN, and give some necessary and sufficient conditions for their boundedness or compactness. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators, for example, M. Lindstrom and E. Wolf: Essential norm of the difference of weighted composition operators. Monatsh. Math. 153 (2008), 133–143.

In this paper, we obtain a symmetry number for the commutator of quasi- homogeneous Toeplitz operators on the harmonic Bergman space. Then we use it to characterize the commuting Toeplitz operators with quasihomogeneous symbols. Also, we show that a Toeplitz operator with an analytic or co-analytic monomial symbol commutes with another Toeplitz operator only in the trivial case.

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 .

In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L1(Bn) symbol.

This paper characterizes the boundedness and compactness of the product of extended Cesàro operator and composition operator from Lipschitz space toF(p,q,s)space on the unit ball ofℂn.

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.

For , let denote the -Bloch space on the unit polydisc of and a holomorphic self-map of . We investigate the boundedness and compactness of the weighted composition between -Bloch space (little -Bloch space ) and -Bloch space (little -Bloch space ). The most important result in the paper is that conditions for the compactness are different for the cases and , unlike for the case of the weighted operators on the unit disc. Bibliography: 32 titles.

In this paper, we first discuss some basic results concerning Toeplitz operators with quasihomogeneous symbols (i.e., symbols being of the form e ipθ ϕ, where ϕ is a radial function) on the harmonic Bergman space. Then we determine when the product of two Toeplitz operators with quasihomogeneous symbols is a Toeplitz operator.

Let g be a holomorphic function of the unit ball B in the n-dimensional complex space, and denote by T g the extended Cesaro operator with symbol g. Starting with a brief introduction to well-known results about Cesaro operator, we investigate the boundedness and compactness of T g from generally weighted Bloch spaces B log α ( 0 α ∞ ) to Zygmund space Z in the unit ball, and also present some necessary and sufficient conditions.

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 .

In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in \({\mathbb{C}}^{n}\). We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form \(\xi^{k} \varphi\) is studied, where \(k \in {\mathbb{Z}}^{n}\) and φ is a radial function.

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].

This paper gives some estimates of the essential norm for the difference of composition operators induced byφandψacting on the space,H∞(Dn), of bounded analytic functions on the unit polydiscDn, whereφandψare holomorphic self-maps ofDn. As a consequence, one obtains conditions in terms of the Carathéodory distance onDnthat characterizes those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators onH∞(Dn)is compact.

In this paper we first look upon some known results on the composition operator as bounded or compact on the Bloch-type space in polydisk and ball, and then give a sufficient and necessary condition for the composition operator to be compact on the Bloch space in a bounded symmetric domain.

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

Let 𝑔 be a holomorphic of the unit ballBin then-dimensional complex space, and denote byTgthe extended Cesáro operator with symbolg. Let 0 −1 and α > 0, starting with a brief introduction to well known results about Cesáro operator, we investigate the boundedness and compactness ofTgbetween generalized Besov spaceB(p, q)and 𝛼α- Bloch spaceℬαin the unit ball, and also present some necessary and sufficient conditions.

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.