Towards a maximal extension of Pełczyński's decomposition method in Banach spaces

Suppose that X, Y, A and B are Banach spaces such that X   is isomorphic to Y⊕AY⊕A and Y   is isomorphic to X⊕BX⊕B. Are X and Y   necessarily isomorphic? In this generality, the answer is no, as proved by W.T. Gowers in 1996. In the present paper, we provide a very simple necessary and sufficient condition on the 10-tuples (k,l,m,n,p,q,r,s,u,v)(k,l,m,n,p,q,r,s,u,v) in NN with p+q+u⩾3p+q+u⩾3, r+s+v⩾3r+s+v⩾3, uv⩾1uv⩾1, (p,q)≠(0,0)(p,q)≠(0,0), (r,s)≠(0,0)(r,s)≠(0,0) and u=1u=1 or v=1v=1 or (p,q)=(1,0)(p,q)=(1,0) or (r,s)=(0,1)(r,s)=(0,1), which guarantees that X is isomorphic to Y whenever these Banach spaces satisfy{Xu∼Xp⊕Yq,Yv∼Xr⊕YsandAk⊕Bl∼Am⊕Bn. Namely, δ=±1δ=±1 or ⋄≠0⋄≠0, gcd(⋄,δ(p+q−u))gcd(⋄,δ(p+q−u)) divides p+q−up+q−u and gcd(⋄,δ(r+s−v))gcd(⋄,δ(r+s−v)) divides r+s−vr+s−v, where δ=k−l−m+nδ=k−l−m+n is the characteristic number of the 4-tuple (k,l,m,n)(k,l,m,n) and ⋄=(p−u)(s−v)−rq⋄=(p−u)(s−v)−rq is the discriminant of the 6-tuple (p,q,r,s,u,v)(p,q,r,s,u,v). We conjecture that this result is in some sense a maximal extension of the classical Pełczyński's decomposition method in Banach spaces: the case (1,0,1,0,2,0,0,2,1,1)(1,0,1,0,2,0,0,2,1,1).