Products of composition and integral type operators from \(H^{\infty}\) to the Bloch space. (English) Zbl 1159.47019

Li, Songxiao; Stević, Stevo

Complex Var. Elliptic Equ. 53, No. 5, 463-474 (2008).

This paper considers a particular type of operators which are the product a composition operator \(C_ \varphi\) and a Volterra-type integral operator \(J_ g\) or, alternatively, of \(C_\varphi\) and another associated integral operator called \(I_ g\). These integral operators are defined as follows: \[ J_ g f(z) = \int_ 0 ^ z f(\xi) g^\prime (\xi) \,d\xi , \qquad I_ g f(z) = \int_ 0 ^ z f^\prime (\xi) g (\xi) \,d\xi, \] for a function \(f\) holomorphic in the unit disk.
In the first part of the paper, the authors characterize those symbols for which the operator \(C_\varphi I_ g\) is bounded or compact from the space of bounded analytic functions into the Bloch space. In the second part, they obtain analogous criteria for the operator \(C_\varphi J_ g\). The paper also contains results for operators into the little Bloch space (in both cases).
Perhaps a relevant paper on the operator \(J_ g\) acting from various Hardy spaces into other spaces by A. Aleman and J. A. Cima [J. Anal. Math. 85, 157–176 (2001; Zbl 1061.30025)], along with some earlier papers by Aleman and Siskakis, could have been cited among the references as well.

Reviewer: Dragan Vukotić (Madrid)

MSC:

47B38	Linear operators on function spaces (general)
30H05	Spaces of bounded analytic functions of one complex variable
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

composition operator; integral operator; Bloch space; bounded analytic functions

Citations:

Zbl 1061.30025

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