On the relationships between \(H^p(\mathbb{T}, X/Y)\) and \(H^p(\mathbb{T},X)/ H^p(\mathbb{T},Y)\) (Q1366585)

First we show that for every \(1\leq p<\infty\) the space \(H^p(\mathbb{T}, L^1(\lambda)/H^1)\) can not 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.