Twisted sums of sequence spaces and the three space problem

Authors:
N. J. Kalton and N. T. Peck

Journal:
Trans. Amer. Math. Soc. **255** (1979), 1-30

MSC:
Primary 46A45

DOI:
https://doi.org/10.1090/S0002-9947-1979-0542869-X

MathSciNet review:
542869

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the following problem: given a complete locally bounded sequence space *Y*, construct a locally bounded space *Z* with a subspace *X* such that both *X* and $Z/X$ are isomorphic to *Y*, and such that *X* is uncomplemented in *Z*. We give a method for constructing *Z* under quite general conditions on *Y*, and we investigate some of the properties of *Z*. In particular, when *Y* is ${l_p} (1 < p < \infty )$, we identify the dual space of *Z*, we study the structure of basic sequences in *Z*, and we study the endomorphisms of *Z* and the projections of *Z* on infinite-dimensional subspaces.

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Additional Information

Keywords:
Locally bounded space,
Banach space,
sequence space,
uncomplemented subspace,
three space problem,
projection,
strictly singular (co-singular) operator

Article copyright:
© Copyright 1979
American Mathematical Society