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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.