## 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.

### 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

Zbl 1061.30025
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### References:

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