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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the existence and uniqueness of complex structure and spaces with ``few'' operators

Author: Stanisław J. Szarek
Journal: Trans. Amer. Math. Soc. 293 (1986), 339-353
MSC: Primary 46B20; Secondary 47D15, 60D05
MathSciNet review: 814926
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Abstract: We construct a $ 2n$-dimensional real normed space whose (Banach-Mazur) distance to the set of spaces admitting complex structure is of order $ {n^{1/2}}$, and two complex $ n$-dimensional normed spaces which are isometric as real spaces, but whose complex Banach-Mazur distance is of order $ n$. Both orders of magnitude are the largest possible. We also construct finite-dimensional spaces with the property that all ``well-bounded'' operators on them are ``rather small'' (in the sense of some ideal norm) perturbations of multiples of identity. We also state some ``metatheorem'', which can be used to produce spaces with various pathological properties.

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Article copyright: © Copyright 1986 American Mathematical Society

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