On extensions of Pełczyński’s decomposition method in 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 . In this paper, we give suitable conditions on finite sums of X and Y to yield that X m is isomorphic to Y n for some m, n ∈ N * . In other words, we obtain some extensions of the well-known Pełczyński decomposition method in Banach spaces. In order to do this, we introduce the notion of Nearly Schroeder-Bernstein Quadruples for Banach spaces and pose a Conjecture to characterise them.

An Arithmetical Characterization of Decomposition Methods in Banach Spaces via Supplemented Subspaces

“…By (8) and (9), it follows that (a) and (b) of Lemma 3.4 are verified. Moreover, it is easy to check that α(1 − r) + β(p − 1) < (q − p)(r − 1) + (p − 1)(s − 1).…”

“…REMARK 2.3. In [8] a quadruple (p, q, r, s) in ‫,ގ‬ with p + q ≥ 2 and r + s ≥ 2 was said to be a Nearly Schroeder-Bernstein Quadruple for Banach spaces (in short, NSBQ) if X m ∼ Y n for some m, n ∈ ‫ގ‬ * whenever the Banach spaces X and Y are isomorphic to complemented subspaces of each other and satisfy the Decomposition Scheme (4 …”

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.…”

Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

“…Moreover, in [6] a quadruple (p, q, r, s) in N, with p + q ≥ 2 and r + s ≥ 2, was said to be a nearly Schroeder-Bernstein quadruple for Banach spaces (in short, NSBQ) if for every pair of Banach spaces X and Y isomorphic to complemeted subspaces of each other and satisfying the decomposition scheme (1.1), we have X m ∼ Y n , for some m, n ∈ N * .…”

“…Our main aim in this paper is to prove [6, Conjecture 1.2] by showing that ∆ = 0 is also a necessary condition to a quadruple (p, q, r, s) in N, with p + q ≥ 2, r + s ≥ 2 to be NSBQ, see Theorem 2.3. In other words, we complete the characterization of the NSBQ conjectured in [6].…”