Towards a maximal extension of Pelczynski`s decomposition method in Banach spaces

l Suppose that X, Y. A and B are Banach spaces such that X is isomorphic to Y E) A and Y is isomorphic to X circle plus B. Are X and Y necessarily isomorphic? In this generality. the answer is no, as proved by W.T. Cowers 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, uv >= 1, (p,q)$(0,0), (r,s)not equal(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) similar to X(p)circle plus Y(q), Y(u) similar to X(r)circle plus Y(s), and A(k) circle plus B(l) similar to A(m) circle plus B(n). Namely, delta = +/- 1 or lozenge not equal 0, gcd(lozenge, delta (p + q - u)) divides p + q - u and gcd(lozenge, delta(r + s - v)) divides r + s - v, where 3 = k - I - in + n is the characteristic number of the 4-tuple (k, l, m, n) and lozenge = (p - u)(s - v) - rq 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 Pelczynski`s decomposition method in Banach spaces: the case (1, 0. 1, 0, 2. 0, 0, 2. 1. 1). (C) 2009 Elsevier Inc. All rights reserved.