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Products of composition and differentiation between Hardy spaces. (English) Zbl 1102.47015
The author discusses the boundedness and compactness of the products of composition and differentiation operators between Hardy spaces, namely, he finds conditions on a symbol $\phi$ in order for the map ${C}_{\phi }Df\left(z\right)={f}^{\text{'}}\left(\phi \left(z\right)\right)$ to define a bounded (or compact) operator from ${H}^{p}$ to ${H}^{q}$, where ${H}^{p}$ stand for the Hardy space on the unit disc $U$. His results cover the cases $1\le p and $2\le p=q<\infty$ and the condition, as expected, becomes a Carleson-type criterion for the measure ${\mu }_{\phi }\left(E\right)={\int }_{{\left(\phi \right)}^{-1}\left(E\right)\cap \partial U\right)}\frac{d\theta }{2\pi }$. In the case $q=\infty$, it is observed that boundedness and compactness are the same and only hold for functions with ${\parallel \phi \parallel }_{\infty }<1$. A more function-theoretic characterization in terms of the Nevanlinna counting function is also provided in the case $p=q=2$, obtaining that ${N}_{\phi }\left(w\right)=O\left(\left[log\left(1/{|w|\right)\right]}^{3}\right)\right]$ and ${N}_{\phi }\left(w\right)=o\left(\left[log\left(1/{|w|\right)\right]}^{3}\right)\right]$ corresponds to the boundedness and compactness, respectively. The proofs use standard techniques and similar results have already been known for weighted Bergman spaces.

MSC:
 47B33 Composition operators 47B38 Operators on function spaces (general) 30H05 Bounded analytic functions
Keywords:
composition operators; Hardy spaces