Decomposition Methods in Banach Spaces via Supplemented Subspaces Resembling Pełczyński’s Decomposition Methods

Abstract

Suppose that X and Y are Banach spaces complemented in each other with supplemented subspaces A and B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y . In this paper, we obtain some suitable conditions involving the spaces A and B to yield that X is isomorphic to Y or to provide that at least X m is isomorphic to Y n for some m, n ∈ IN * . So we get some decomposition methods in Banach spaces via supplemented subspaces resembling Pe lczyński's decomposition methods. In order to do this, we introduce several notions of Schroeder-Bernstein Quadruples acting on the spaces X, Y , A and B. Thus, we characterize them by using some Banach spaces recently constructed.

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

“…So by the main result of [9] mentioned in the introduction, X ∼ Y . Furthermore, if δ = 1, then as we already said in the introduction, [10,Proposition 3.2] implies that X ∼ Y . When δ 2 we distinguish four cases: s = 0; s = 1; s = 2; s 3.…”

“…Gowers [14] solved the so-called Schroeder-Bernstein Problem for Banach spaces by showing that even under this condition, X and Y need not to be isomorphic, see also [3][4][5][6][7][8]15]. So in the study of the geometry of Banach spaces when we have the situation (1.1) we search for additional conditions on X , Y , A and B to yield X ∼ Y , see for example [10][11][12][13]. This area of research originated from the Pełczyński's works.…”

“…According to [10,Proposition 5.6] we see that a 10-tuple is an SBT whenever δ = ±1. Furthermore, [9, Theorem 1.2] implies that a 10-tuple (1, 0, 1, 0, p, q, r, s, 1, v) in N is an SBT whenever ♦ = 0, ♦ divides p + q − u and r + s − v. In the present paper we unify and extend these results working towards a maximal extension of Pełczyński's decomposition method in Banach spaces.…”

Generalizations of Pełczyński’s decomposition method for Banach spaces containing a complemented copy of their squares

“…In this section is convenient to recall, see [7], that the characteristic number of a quadruple (p, q, r, s) in N is given by δ = p−q −r +s. Notice that the characteristic number of (r, s, p, q) is r − s − p + q = −δ.…”