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 ∞. 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 (ℓ q 0), H p 1 (ℓ q 1), with 1 ≤ p 0 , p 1 , q 0 , q 1 ≤ ∞. 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 (ℓ q 0), L p 1 (ℓ q 1). In an other direction, let us denote by C p the space of all compact operators x on Hilbert space such that tr(|x| p) < ∞. Let T p be the subspace of all upper triangular matrices relative to the canonical basis. If p = ∞, 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 ∞ (C ∞) (resp. T ∞). 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 ∞ can be essentially realized simultaneously by the same element.