TY - JOUR
T1 - Smooth noncompact operators from C(K), K scattered
AU - Deville, R.
AU - Hájek, P.
PY - 2007/12
Y1 - 2007/12
N2 - 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
c0ℳ 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.
AB - 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
c0ℳ 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.
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U2 - 10.1007/s11856-007-0086-7
DO - 10.1007/s11856-007-0086-7
M3 - Article
SN - 0021-2172
VL - 162
SP - 29
EP - 56
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -