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Generalized composition operators on Zygmund spaces and Bloch type spaces. (English) Zbl 1135.47021

Let D denote the unit disc in the complex plane, let H(D) denote the set of all functions holomorphic on D, and let C(D ¯) denote the set of all functions continuous on the closure of D. A Bloch type space (or α-Bloch space) is a space of the form

B α ={fH(D):sup zD (1-|z| 2 ) α |f ' (z)|<},

where the space B α is given the norm

f B α =|f(0)|+sup zD (1-|z| 2 ) α |f ' (z)|·

The Zygmund space Z is the space

Z=fH(D)C(D ¯):sup θ[0,2π],h>0 |f(e iθ+h )+f(e iθ-h )-2f(e iθ )| h<,

with the norm given by

f Z =|f(0)|+|f ' (0)|+sup zD (1-|z| 2 )|f '' (z)|·

Throughout, ϕ denotes a non-constant analytic self-map of D. A basic composition operator is given by C ϕ f=fϕ for fH(D). Let gH(D) and define the linear operator

(C ϕ g f)(z)= 0 z f ' (ϕ(ζ))g(ζ)dζ·

The authors give criteria under which the general composition operator C ϕ g :ZB α is a bounded operator, and also when it is a compact operator. Also considered are the cases when C ϕ g :ZZ and when C ϕ g :B α Z is a bounded operator, and when it is a compact operator. Letting

B 0 α =fB α :lim |z|1 (1-|z| 2 ) α |f ' (z)=0,

and letting

Z 0 =fZ:lim |z|1 (1-|z| 2 )|f '' (z)|=0,

corresponding results are obtained using B 0 α in place of B α and using Z 0 in place of Z. Two typical results are as follows. Theorem. If 0<α<, if gH(D) and ϕ is an analytic self-map of D, then C ϕ g :ZB α is bounded if and only if

sup zD (1-|z| 2 ) α |g(z)|log1 1-|ϕ(z)| 2 <·

In addition, this operator is compact if and only it is bounded and

lim |ϕ(z)|1 (1-|z| 2 ) α |g(z)|log1 1-|ϕ(z)| 2 =0·

Theorem. If 0<α<, if gH(D), and if ϕ is an analytic self-map of D, then the following statements are equivalent: (i) C ϕ g :B α Z is compact; (ii) C ϕ g :B 0 α Z is compact; (iii) C ϕ g :B α Z is bounded and both

lim |ϕ(z)|1 (1-|z| 2 )|ϕ ' (z)||g(z)| (1-|z| 2 ) α+1 =0andlim |ϕ(z)|1 (1-|z| 2 )|g ' (z)| (1-|ϕ(z)| 2 ) α =0·


MSC:
47B33Composition operators
30D45Bloch functions, normal functions, normal families
47B38Operators on function spaces (general)
46E15Banach spaces of continuous, differentiable or analytic functions
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