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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]