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

Li, Songxiao; Stević, Stevo

J. Math. Anal. Appl. 349, No. 2, 596-610 (2009).

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.

Reviewer: Elke Wolf (Paderborn)

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

Citations:

Zbl 0369.30012

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