Weak Grothendieck's theorem

Let En ⊂ L 1 be the n-dimensional subspace which appeared in Kašin’s theorem such that L 1 = En ⊕E⊥ n and the L 1 and L 2 norms are universally equivalent on both En and E⊥ n . In this paper, we introduce and study some properties concerning extension and weak Grothendieck’s theorem (WGT). We show that the Schatten space Sp for all 0 < p ≤ ∞ does not verify the theorem of extension. We prove also that Sp fails GT for all 1 ≤ p ≤ ∞ and consequently by one result of Maurey does not satisfy WGT for 1 ≤ p ≤ 2. We conclude by giving a characterization for spaces verifying WGT.