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Let 𝑔 be a holomorphic of the unit ball B in the n-dimensional complex space, and denote by Tg the extended Cesáro operator with symbol g. Let 0 < p < +∞, −n − 1 < q < +∞, q > −1 and α > 0, starting with a brief introduction to well known results about Cesáro operator, we investigate the boundedness and compactness of Tg between generalized Besov space B(p, q) and 𝛼α- Bloch space ℬα in the unit ball, and also present some necessary and sufficient conditions.

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 characterize the boundedness and compactness of differences of the composition operators followed by differentiation between weighted Banach spaces of holomorphic functions in the unit disk. As their corollaries, some related results on the differences of composition operators acting from weighted Banach spaces to weighted Bloch type spaces are also obtained.

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

Let Bn be the unit ball of ℂn and ϕ=(ϕ1,…,ϕn) a holomorphic self-map of Bn. Let 0-1, α>0, and let Cϕ be the composition operator between the space F(p,q,s) and α-Bloch space ℬα induced by ϕ. This paper gives an estimate of the essential norm of Cϕ. As a consequence, a necessary and sufficient condition for the composition operator Cϕ to be compact from F(p,q,s) to ℬα is obtained.

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

Let BN be the unit ball in the N-dimensional complex space, for È, a holomorphic function in BN, and Õ, a holomorphic map from BN into itself, the weighted composition operator on the weighted Hardy space H2(β,BN) is given by (CÈ,Õ)f=È(z)f(Õ(z)), where f∈H2(β,BN). This paper discusses the spectrum of CÈ,Õ when it is compact on a certain class of weighted Hardy spaces and when the composition map Õ has only one fixed point inside the unit ball.

In this paper, we characterize the boundedness and compactness for weighted composition operators on the Fock-Sobolev space F p,m(Cn), 0 < p < ∞. We prove that no nontrivial unitary or self-adjoint weighted composition operators exist on F2,m(Cn) when m ≥ 1. As an application, we also prove that there exist only trivial J -symmetric weighted composition operators on F2,m(Cn) when m ≥ 1. [ABSTRACT FROM AUTHOR]

In this article, we give equivalent conditions for the hypercyclicity of bilateral operator-weighted shifts on L²(K) with weight sequence {An}n∞ =-∞ of positive invertible diagonal operators on a separable complex Hilbert space K, as well as for hereditarily hypercyclicity and supercyclicity. [ABSTRACT FROM AUTHOR]

We study how certain Volterra type operators compactly intertwine with composition operators on the space of bounded analytic functions and composition operators on the Bloch space of the unit disk. [ABSTRACT FROM AUTHOR]

Let φ denote a nonconstant analytic self-map of the open unit disk D, g be an analytic function on D. In this paper, we characterize the necessary or sufficient conditions for generalized composition operators Cgφ f(z) = ∫ z 0 f'(φ(ξ))g(ξ)dξ, on the Bloch-type spaces to have a closed range. Moreover, if g ∊ H∞, according to relationship betweenα and β, we show several conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we use Young's function to define Zygmund-Orlicz space. We study boundedness and compactness of composition operator on Zygmund-Orlicz space. [ABSTRACT FROM AUTHOR]

Some sufficient conditions under which the tuple of the adjoint of weighted composition operators (C*ω1,φ1, C*ω2,φ2) on the Hilbert space ℋ of analytic functions is disk-cyclic (or codisk-cyclic) were investigated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space H²(BN) and weighted Bergman spaces Aa²(BN), where BN is the unit ball of the N-dimensional complex space. By taking N = 1, BN = D the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on H²(D) or A²(D). [ABSTRACT FROM AUTHOR]

A bounded linear operator T on a Banach space X is called subspace-hypercyclic for a nonzero subspace M if orb (T, x) ∩ M is dense in M for a vector x ∊ X, where orb(T, x) = {Tnx : n = 0, 1, 2, ... }. 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 subspacesupercyclic operators. [ABSTRACT FROM AUTHOR]

We characterize the disjoint supercyclicity of finitely many different powers of weighted shifts acting on the weighted sequence spaces l²(...,w), c0(...,w), and l²(Z,w), c0(...,w), where w = (w1)1 is a positive weight sequence satisfying w1 ≥ 1 for every i ∈ ... (or i ∈ ...). [ABSTRACT FROM AUTHOR]

Let S(픻) be the collection of all the holomorphic self-maps of open unit disk 픻 of the complex plane ℂ, and Cφ, the composition operator induced by φ ∈ S(픻). For a > 0, λ ∈ ℂ, we give some sufficient and necessary conditions for the hypercyclicity of multiples of composition operators #955;Cφ acting on the weighted Banach spaces of entire functions H∞α,0. Moreover, we obtain a partial characterization for the frequent hypercyclicity of #955;Cφ on H∞α,0. [ABSTRACT FROM AUTHOR]

We characterize when N-tuples Br1a1, ⋯, BrNaNof powers of unilateral weighted back-ward shift operators defined the space of all complex valued square summable sequences exhibit the d-mixing properties, and give several equivalent conditions of d-mixing properties. At the same time, d-mixing powers of bilateral weighted backward shifts on Banach sequence spaces are also discussed in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the isometries of the products of composition and differentiation operators from the Bloch type space to the space of all weighted bounded analytic functions in the disk. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the isometries of the products of composition and differentiation operators from the Bloch type space to the space of all weighted bounded analytic functions in the disk. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the boundedness and compactness of the product of extended Cesàro operator and composition operator from the logarithmic Bloch-type space to F(p, q, s) space on the unit ball of ℂn. [ABSTRACT FROM AUTHOR]

For φ a holomorphic self-map of the unit ball BN of ℂN, and u ∈ H∞(BN) (the Banach space of bounded holomorphic functions on BN), we investigate the essential norm and spectrum of the weighted composition operator uCφ acting on the space H∞(BN). For φ univalent, not unitary on any slice, and fixing a point of BN, we obtain a complete characterization of the spectrum of uCφ. [ABSTRACT FROM AUTHOR]

For φ a holomorphic self-map of the unit ball BN of ℂN, and u ∈ H∞(BN) (the Banach space of bounded holomorphic functions on BN), we investigate the essential norm and spectrum of the weighted composition operator uCφ acting on the space H∞(BN). For φ univalent, not unitary on any slice, and fixing a point of BN, we obtain a complete characterization of the spectrum of uCφ. [ABSTRACT FROM AUTHOR]

Let φ(z) = (φ1 (z), … , φn(z)) be a holomorphic self-map of Bn and H(Bn) a holomorphic function on Bn,, and H(Bn) the class of all holomorphic functions on Bn, where Bn is the unit ball of Cm, the weight composition operator Wψφ is defined by Wφ,φ = ψƒ(φ) for ƒ ϵ H(Bn). In this paper we estimate the essential norm for the weighted composition operator Wφ,φ acting from the Hardy space Hp to Hq (0 < p,q ≤ ∞). When p = ∞ and q = 2, we give an exact formula for the essential norm. As their applications, we also obtain some sufficient and necessary conditions for the bounded weighted composition operator to be compact from Hp to Hq. [ABSTRACT FROM AUTHOR]

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class Sp are obtained. We also construct a compact composition operator which does not belong to any Schatten class. [ABSTRACT FROM AUTHOR]

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class Sp are obtained. We also construct a compact composition operator which does not belong to any Schatten class. [ABSTRACT FROM AUTHOR]

In 1987, Shapiro showed that composition operator induced by symbol φ is compact on the Lipschitz space if and only if the infinity norm of φ is less than 1 by a spectral-theoretic argument, where φ is a holomorphic self-map of the unit disk. In this note, we shall generalize Shapiro's result to the n-dimensional case. [ABSTRACT FROM AUTHOR]

Let Bn be the unit ball of Cn and φ = (φ1, . . . , φn) a holomorphic self-map of Bn. Let 0 < p,s < ∞, −n−1 < q < ∞, q s > −1,α > 0, and let Cφ be the composition operator between the space F(p, q, s) and α-Bloch space Bα induced by φ. This paper gives an estimate of the essential norm of Cφ. As a consequence, a necessary and sufficient condition for the composition operator Cφ to be compact from F(p, q, s) to Bα is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we first discuss some basic results concerning Toeplitz operators with quasihomogeneous symbols (i.e., symbols being of the form $ e^{iptheta }varphi $, where $varphi $ 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the compactness of linear combinations of composition operators acting on bounded holomorphic function space H∞(BN) in the unit ball of CN, and completely characterize the association of compactness and coefficients of linear combinations of composition operators. [ABSTRACT FROM AUTHOR]

The article reports on the study of the spectrum structure of compact operators on Hardy spaces of a unit ball. It aims to determine the spectrum of compact operators in weighted hardy spaces of a unit ball through eigenvalues of the compact operators. It uses differential equations and different theorems of inequalities in analyzing the spectrum structure. It also shows that the spectral radius of compact operators are nonautomorphism linear fractional map which takes the unit sphere to an ellipsoid sphere.

In this article, we first give an essential characterization of Toeplitz operators with quasihomogeneous symbols on the weighted pluriharmonic Bergman space of the unit polydisk. Then we completely characterize when the product of two Toeplitz operators with monomial-type symbols is a Toeplitz operator. As a result, some interesting higher-dimensional phenomena appear on the unit polydisk. [ABSTRACT FROM AUTHOR]

In this paper, we characterize nth differentiation composition operators from weighted Banach space of holomorphic function to weighted Bloch space, and give some necessary and sufficient conditions for the boundedness and compactness of the operators. [ABSTRACT FROM AUTHOR]