## Products of integral-type operators and composition operators between Bloch-type spaces.(English)Zbl 1155.47036

In [Comment. Math. Helv. 52, 591-602 (1977; Zbl 0369.30012)], Ch. Pommerenke introduced the following integral type operator
$J_g f(z) = \int_0^z f(\xi) g'(\xi) \; d \xi$
for $$z \in \mathbb D$$, a holomorphic map $$g: \mathbb D \to \mathbb C$$ and $$f \in H(\mathbb D)$$. The integral type operator
$I_g f(z) = \int_0^z f'(\xi) g(\xi) \; d \xi, \quad z \in \mathbb D,$
is closely connected to $$J_g$$ by the inequality
$J_g f + I_g f = M_g f - f(0) g(0),$
where $$M_g$$ is the multiplication operator given by
$(M_g f)(z)= g(z) f(z).$
For an analytic self-map $$\varphi$$ of $$\mathbb D$$, the composition operator is defined by $$C_{\varphi} f= f \circ \varphi$$ for $$f \in H(\mathbb D)$$. This paper is devoted to the study of the following products of composition operators and integral-type operators
\begin{aligned} C_{\varphi} J_g(f)(z) &= \int_0^{\varphi(z)} f(\xi) g'(\xi) \,d \xi,\\ C_{\varphi} I_g(f)(z) &= \int_0^{\varphi(z)} f'(\xi) g(\xi) \, d \xi,\\ J_g C_{\varphi}(f)(z) &= \int_0^z (f \circ \varphi) (\xi) g'(\xi) \,d \xi,\\ I_g C_{\varphi}(f)(z) &= \int_0^z (f \circ \varphi)' (\xi) g(\xi) \, d \xi. \end{aligned} The authors characterize when such products acting between the $$\alpha$$-Bloch spaces
$B^{\alpha} = \{f \in H(\mathbb D); \quad \| f\| _{B^{\alpha}} = | f(0)| + \sup_{z \in \mathbb D} (1-| z| ^2)^{\alpha} | f'(z)| < \infty\},\quad \alpha > 0,$
resp., the little $$\alpha$$-Bloch spaces
$B_0^{\alpha}= \Bigl\{ f \in B^{\alpha}; \; \lim_{| z| \to 1} (1-| z| ^2)^{\alpha} | f'(z)| =0 \Bigr\},\quad \alpha > 0,$
are bounded, resp. compact.

### MSC:

 47B38 Linear operators on function spaces (general) 30H05 Spaces of bounded analytic functions of one complex variable

### Keywords:

integral operator; composition operator; Bloch space

Zbl 0369.30012
Full Text:

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