Towards a maximal extension of Pe?czyński's decomposition method in Banach spaces

摘要

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 {white diamond suit} ≠ 0, gcd ({white diamond suit}, δ (p + q - u)) divides p + q - u and gcd ({white diamond suit}, 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 {white diamond suit} = (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 Pe?czyński's decomposition method in Banach spaces: the case (1, 0, 1, 0, 2, 0, 0, 2, 1, 1).