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

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.


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


Zbl 1061.30025
Full Text: DOI


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