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...

The aim of the paper is to propose a definition of numerical range of an operator on reflexive Banach spaces. Under this definition the numerical range will possess the basic properties of a canonical numerical range. We will determine necessary and sufficient conditions under which the numerical range of a composition operator on a weighted Hardy space is closed. We will also give some necessary conditions to show that when the closure of the numerical range of a composition operator on a small...