A property of interpolation spaces

Let Ao, A~ and A 2 be Banach spaces continuously imbedded in some Hausdorff topological vector space, and let F be an interpolation functor. We consider the question: when is it true that (1) F({Ao, A 1 c~ A2} ) = F({Ao, A1} ) n F({Ao, A2) ). Peetre [4] proved that if {Ao, A~} is quasi-linearizable pair, i.e., there exist linear operators Vo(t ), Vl(t ) (depending on t>0) such that: VI(t):Ao+A~A i, i=0,1, Vo(t)a + V~(t)a = aand I[ Vo(t)allao + t II V~(t)aflA1 < cK(t, a: A o, A0 for a sA o + A1, and if moreover LlVl(t)alta <=c21lalla~ for a~A 2, then for a ~ (A o + A 0 c~ (A o + A2) , we have (2)