# On the relationships between $H^p(\mathbb{T} ,X/Y)$ and $H^p(\mathbb{T} ,X)/H^p(\mathbb{T} ,Y)$

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Universitat de Barcelona
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## Abstract

First we show that for every $1\leq p <\infty$ the space $H^p(\mathbb{T}, L^1(\lambda)/H^1)$ cannot be naturally identified with $H^p(\mathbb{T}, L^1(\lambda))/H^p(\mathbb{T},H^1)$. Next we show that if $Y$ is a closed locally complemented subspace of a complex Banach space $X$ and $0 < p <\infty$, then the space $H^p(\mathbb{T},X/Y )$ is isomorphic to the quotient space $H^p(\mathbb{T},X)/H^p(\mathbb{T}, Y )$. This allows us to show that all odd duals of the James Tree space $JT_2$ have the analytic Radon-Nikodym property.

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