On products of noncommutative symmetric quasi Banach spaces and applications

Abstract

Let E 1 , E 2 be symmetric quasi Banach spaces on [0, α) (0 < α ≤ ∞). We collected and proved some properties of the space E 1 ⊙ E 2 , where ⊙ means the pointwise product of symmetric quasi Banach spaces. Under some natural assumptions, weAs application, we extend this results to the noncommutative symmetric quasi spaces and the noncommutative symmetric quasi Hardy spaces case. We also obtained the real case of Peter Jones' theorem for noncommutative symmetric quasi Hardy spaces. Keywords:Symmetric quasi Banach space, pointwise product of symmetric quasi Banach spaces, Noncommutative symmetric quasi space, Noncommutative symmetric quasi Hardy space, complex and real interpolation. 46L52, 47L51

Interpolation of Haagerup noncommutative Hardy spaces

“…in the sense of equivalence of quasi norms (see [3,Theorem 5]). Since L p 0 ,∞ 0 (0, ∞) + L p 1 ,∞ 0 (0, ∞) is separable and s-convex (s = min{s 1 , s 2 }), we use the method in the proof of [3,Lemma 1] to obtain that there is an equivalent quasi norm • on L p 0 ,∞ 0 (N ) + L p 1 ,∞ 0 (N ) which is plurisubharmonic. According to [33,Lemma 4.5], the separability and s-convexity of L p 0 ,∞ 0 (0, ∞)…”