Abstract :
Suppose that X, Y, A and B are Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ 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 ) in N with p + q + u ⩾ 3 , r + s + v ⩾ 3 , u v ⩾ 1 , ( p , q ) ≠ ( 0 , 0 ) , ( r , s ) ≠ ( 0 , 0 ) and u = 1 or v = 1 or ( p , q ) = ( 1 , 0 ) or ( r , s ) = ( 0 , 1 ) , which guarantees that X is isomorphic to Y whenever these Banach spaces satisfy { X u ∼ X p ⊕ Y q , Y v ∼ X r ⊕ Y s and A k ⊕ B l ∼ A m ⊕ B n . Namely, δ = ± 1 or ⋄ ≠ 0 , gcd ( ⋄ , δ ( p + q − u ) ) divides p + q − u and gcd ( ⋄ , δ ( r + s − v ) ) divides r + s − v , where δ = k − l − m + n is the characteristic number of the 4-tuple ( k , l , m , n ) and ⋄ = ( p − u ) ( s − v ) − r q is the discriminant of the 6-tuple ( p , q , r , s , u , v ) . We conjecture that this result is in some sense a maximal extension of the classical Pełczyńskiʹs decomposition method in Banach spaces: the case ( 1 , 0 , 1 , 0 , 2 , 0 , 0 , 2 , 1 , 1 ) .
