Interpolation between H^p spaces and non-commutative generalizations, I

We give an elementary proof that the $H^p$ spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between $H^1$ and $H^\infty$. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the classical factorisation of a function in $H^p$ as a product of two functions in $H^q$ and $H^r$ with $1/q+1/r=1/p$. This proof extends without any real extra difficulty to the non-commutative setting and to several Banach space valued extensions of $H^p$ spaces. In particular, this proof easily extends to the couple $H^{p_0}(\ell_{q_0}),H^{p_1}(\ell_{q_1})$, with $1\leq p_0, p_1, q_0, q_1 \leq \infty$. In that situation, we prove that the real interpolation spaces and the K-functional are induced ( up to equivalence of norms ) by the same objects for the couple $L_{p_0}(\ell_{q_0}), L_{p_1}(\ell_{q_1})$. In another direction, let us denote by $C_p$ the space of all compact operators $x$ on Hilbert space such that $tr(|x|^p) <\infty$. Let $T_p$ be the subspace of all upper triangular matrices relative to the canonical basis. If $p=\infty$, $C_p$ is just the space of all compact operators. Our proof allows us to show for instance that the space $H^p(C_p)$ (resp. $T_p$) is the interpolation space of parameter $(1/p,p)$ between $H^1(C_1)$ (resp. $T_1$) and $H^\infty(C_\infty)$ (resp. $T_\i$). We also prove a similar result for the complex interpolation method. Moreover, extending a recent result of Kaftal-Larson and Weiss, we prove that the distance to the subspace of upper triangular matrices in $C_1$ and $C_\infty$ can be essentially realized simultaneously by the same element.