Composition Operators fronm Weak to Strong Vector-Valued Hardy Spaces

Abstract

Let $$\phi $$ be an analytic map from the unit disk into itself, $$1<p<2$$ and $$1\le q \le p$$ . It is shown that the composition operator $$C_\phi (f)=f\circ \phi $$ is bounded from $$H^p_{weak}({\mathbb D},L^q(\mu ))$$ into $$H^p({\mathbb D},L^q(\mu ))$$ if and only if $$C_\phi $$ is a 2-summing operator from $$H^p({\mathbb D})$$ into $$H^p({\mathbb D})$$ . Here $$H^p_{weak}({\mathbb D},X)$$ and $$H^p({\mathbb D},X)$$ are the weak and strong formulation of X-valued Hardy spaces on the unit disc.