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On Banach spaces, nonisomorphic to their Cartesian squares. (English. Russian original) Zbl 0844.46011
Math. Notes 57, No. 4, 369-374 (1995); translation from Mat. Zametki 57, No. 4, 534-541 (1995).
The first examples of Banach spaces, nonisomorphic to their Cartesian squares, have been given by C. Bessaga, A. Pełczyński [Bull. Acad. Polon. Sci. 8, No. 2, 77-80 (1960; Zbl 0091.27801)] and Z. Semadeni [ibid. 8, No. 2, 81-84 (1960; Zbl 0091.27802)]. An example of a reflexive space with this property has been constructed by T. Figiel [Studia Math. 42, No. 3, 295-306 (1972; Zbl 0213.12801)]. As shown by S. Szarek [Proc. Am. Math. Soc. 97, No. 3, 437-444 (1986; Zbl 0604.46019)], there exists a real Banach space that is not isomorphic to the Cartesian square of any Banach space. The fundamental result of the present paper is the following
Theorem. There exist reflexive Banach spaces \(X\) and \(Y\), nonisomorphic to their Cartesian squares but having the property that \(X\oplus Y\) is isomorphic to its Cartesian square.
46B20 Geometry and structure of normed linear spaces
Full Text: DOI
[1] C. Bessaga and A. Pełczyński, ”Banach spaces non-isomorphic to their Cartesian squares. I,” Bull. Acad. Polon. Sci.,8, No. 2, 77–80 (1960). · Zbl 0091.27801
[2] Z. Semadeni, ”Banach spaces non-isomorphic to their Cartesian squares. II,” Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.,8, No. 2, 81–86 (1960). · Zbl 0091.27802
[3] T. Figiel, ”An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square,” Studia Math.,42, No. 3, 295–306 (1972). · Zbl 0213.12801
[4] S. J. Szarek, ”A superreflexive Banach space which does not admit complex structure,” Proc. Amer. Math. Soc.,97, No. 3, 437–444 (1986). · Zbl 0604.46019
[5] T. Figiel, J. Lindenstrauss, and V. D. Milman, ”The dimension of almost spherical sections of convex bodies,” Acta Math.,139, No. 1–2, 53–94 (1977). · Zbl 0375.52002
[6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Function Spaces, Springer, Berlin (1979). · Zbl 0403.46022
[7] V. M. Kadets and K. E. Kaibkhanov, ”On the structure of sets of admissible perturbations,” Teor. Funktsii Funktsional. Anal. i Prilozhen. (Khar’kov), No. 53, 79–87 (1990). · Zbl 0784.47023
[8] S. Kwapien, ”Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients,” Studia Math.,44, 583–595 (1972). · Zbl 0256.46024
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