Abstract
We obtain a criterion for the boundedness and compactness of the products of differentiation and composition operators C φ D m on the logarithmic Bloch space in terms of the sequence { z n }. An estimate for the essential norm of C φ D m is given.MSC:47B38, 30H30.
Highlights
Denote by H(D) the space of all analytic functions on the unit disk D = {z : |z| < } in the complex plane
It is well known that LB ∩ H∞ is the space of multipliers of the Bloch space B
A basic problem concerning concrete operators on various Banach spaces is to relate the operator theoretic properties of the operators to the function theoretic properties of their symbols, which attracted a lot of attention recently, the reader can refer to [ – ]
Summary
A basic problem concerning concrete operators on various Banach spaces is to relate the operator theoretic properties of the operators to the function theoretic properties of their symbols, which attracted a lot of attention recently, the reader can refer to [ – ]. It is a well-known consequence of the Schwarz-Pick lemma that the composition operator is bounded on B. In [ ], Wu and Wulan obtained a characterization for the compactness of the product of differentiation and composition operators acting on the Bloch space as follows: Theorem A Let φ be an analytic self-map of D, m ∈ N. CφDm : B → B is compact if and only if lim n→∞
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