Abstract :
Let X be a uniformly smooth infinite dimensional Banach space, and (Ω,Σ,μ) be a σ -finite
measure space. Suppose that T :X→L∞(Ω,Σ,μ) satisfies
(1 − ε) x T x x , ∀x ∈ X,
for some positive number ε <1/2 with δX∗ (2−2ε) > 13/14. Then T is close to an isometry U :X→
L∞(Ω,Σ,μ) such that
T − U 16 1 − δX∗ (2 −2ε) +
1
2
ε,
where δX∗ (t ) is the modulus of convexity of the conjugate space X∗.
2003 Elsevier Inc. All rights reserved
