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The composition operators on weighted Bloch space. (English) Zbl 1038.47020

Let H(D) denote the set of all holomorphic functions in the open unit disk D and let ϕ be a holomorphic selfmap of D. Suppose that X and Y are two function spaces defined on D. The map ϕ is said to have the X-to-Y pullback property if fϕY for every fX. Let I be an arc on the unit circle and S(I)={re iθ :1-r|I|, e iθ I} the associated Carleson box. In the present paper, the author studies the X-to-Y pullback property for

X=Y=B log :=fH(D)|sup zD (1-|z| 2 )log2 1-|z| 2 |f ' (z)|<,

for

X=Y=BMOA log :=fH(D)|sup I log2 |I| 2 |I| S(I) |f ' (z)| 2 log(1/|z|)dA(z)<,

and for X=B log and Y=BMOA log . A complete characterization of the bounded and compact composition operators C ϕ :ffϕ on the weighted Bloch space B log is given. Unfortunately, this paper has many grammatical errors and the title of reference [1] is not correct.


MSC:
47B33Composition operators
30H05Bounded analytic functions