The Schroeder–Bernstein index for Banach spaces

2007

Result. Math.

Self Cite

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.

Section: Decomposition Methods In Banach Spaces Via Supplemented Subsmentioning

confidence: 99%

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

2007

Result. Math.

Self Cite

“…Remark 2.1. Fix two Banach spaces X and Y from the class of spaces constructed in [7], as was done in [4]. By [4] there exists a Banach space Z such that …”

Section: Characterization Of Nearly Schroeder-bernstein Quadruples Fomentioning

confidence: 99%

Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

2006

Arch. Math.

Self Cite

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.

“…Thus r = 0. Now fix two Banach spaces X and Y from the class of spaces constructed in [6], as was done in [4]. By [4] there exist Banach spaces Z and E isomorphic to complemented subspaces of each other such that …”

Section: On Quadruples Inmentioning

confidence: 99%

On extensions of Pełczyński’s decomposition method in Banach spaces

2005

Arch. Math.

Self Cite

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.