In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space.

Let $\varphi$ and $\psi$ be holomorphic self-maps of the unit disk, and denote by $C_{\varphi }$, $C_{\psi }$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_{\varphi }-C_{\psi }$ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.

Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u{C}_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$, $\varphi$, their derivatives, and ${\varphi }^n$, the $n$-th power of $\varphi$. Moreover, we obtain some similar estimates for the essential norms of the operators $u{C}_{\varphi }$, from which sufficient and necessary conditions of compactness of $u{C}_{\varphi }$ follows immediately.

Let ψ and φ be analytic functions on the open unit disk $𝔻$ with φ($𝔻$) ⊆ $𝔻$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $𝒟$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations...

Let $\varphi$ be an analytic self-mapping of $𝔻$ and $g$ an analytic function on $𝔻$. In this paper we characterize the bounded and compact Volterra composition operators from the Bergman-type space to the Bloch-type space. We also obtain an asymptotical expression of the essential norm of these operators in terms of the symbols $g$ and $\varphi$.

Let ϕ: → and ψ: → ℂ be analytic maps. They induce a weighted composition operator $\psi C_{\varphi }$ acting between weighted Bergman spaces of infinite order and weighted Bloch type spaces. Under some assumptions on the weights we give a characterization for such an operator to be bounded in terms of the weights involved as well as the functions ψ and ϕ