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A new characterization of boundedness and compactness for differences of differentiation composition operators between Bloch spaces. (English) Zbl 1438.47060

Summary: Let \(\varphi \) and \(\psi \) be analytic self-maps of the open unit disk \(D\) in the complex plane \(\mathbb{C}\). The operator \({C_\varphi}{D^n}\), which maps an analytic function \(f\) to \({f^{\left(n \right)}} \circ \varphi \), is called differentiation composition operator, where \({f^{\left(n \right)}}\) denotes the \(n\)-th derivative of \(f\). In this paper, we give some new characterizations of the boundedness and compactness for the differences of differentiation composition operators \({C_\varphi}{D^n} - {C_\psi}{D^n}\) from the Bloch space to the Bloch space in the open unit disk \(D\). Some estimates for the essential norm of differences of differentiation composition operators \({C_\varphi}{D^n} - {C_\psi}{D^n}\) between Bloch spaces in the open unit disk \(D\) are also considered.

MSC:

47B38 Linear operators on function spaces (general)
47B33 Linear composition operators
30H30 Bloch spaces
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