## Abstract

Let X be a Banach space, K be a scattered compact and T: B
_{C(K)} → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B
_{C(K)**} → X**and prove that if T is noncompact, then the derivative of T**at some point is a noncompact linear operator. Using this we conclude, among other things, that either T(B
_{c0}ℳ is compact or that ℓ
_{1} is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C
^{1,u} -smooth noncompact operator from B
_{c}
_{O} which does not fix any (affine) basic sequence.

Original language | English (US) |
---|---|

Pages (from-to) | 29-56 |

Number of pages | 28 |

Journal | Israel Journal of Mathematics |

Volume | 162 |

DOIs | |

State | Published - Dec 2007 |

## ASJC Scopus subject areas

- Mathematics(all)