The essential norm of a composition operator on Bloch spaces. (English) Zbl 0932.30034

A function \(f\) analytic on the unit disk \(D\), is said to belong to the Bloch space \({\mathcal B}\) if \(\sup_D(1- |z|^2)|f'(z) |< \infty\). The author shows the following: When \(C_\varphi\) is a composition operator on \({\mathcal B}\) and \(\|C_\varphi\|_e\) is an essential norm, \(\|C_\varphi \|_e= \lim_{s\to 1^-} \sup_{|\varphi(z)|>s}{1-|z|^2\over 1-|\varphi(z)|^2} |\varphi' (z)|\). This beautiful formula is also true for the little Bloch space \({\mathcal B}_0\) and he obtains a new proof of a recently obtained characterization of the compact composition operators on Bloch spaces.
Reviewer: T.Nakazi (Sapporo)


30D45 Normal functions of one complex variable, normal families
47B38 Linear operators on function spaces (general)
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