Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

Abstract

Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y . Let (p, q, r, s) be a quadruple in N with p + q ≥ 2 and r + s ≥ 2. Suppose that for every pair of Banach spaces X and Y isomorphic to complemented subspaces of each other and satisfying the following Decomposition Schemewe conclude that X m is isomorphic to Y n for some m, n ∈ N * . In this paper, we show that the discriminant ∆ = (p − 1)(s − 1) − rq of this quadruple is different from zero. This result completes the characterization of quadruples in N which are nearly Schroeder-Bernstein Quadruples for Banach spaces. Mathematics Subject Classification (2000). Primary 46B03, 46B20.

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

“…[14, page 563] will be fundamental in the proof of this characterization, see Remark 2.1 and the proofs of Lemmas 3.1 and 3.3. Next inspired by the results of [9] and [11] above mentioned, we also define: In fourth section we use two Banach spaces constructed in [5], see Remark 2.2, to obtain the following characterization of the quadruples in IN which are NSBQS. Theorem 1.5.…”