On stability of nonlinear non-surjective ε-isometries of Banach spaces

By Lixin Cheng, Y. Dong, Wen Zhang. 2013

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Abstract Let X , Y be two Banach spaces, e ⩾ 0 , and let f : X → Y be an e -isometry with f ( 0 ) = 0 . In this paper, we show first that for every x ⁎ ∈ X ⁎ , there exists ϕ ∈ Y ⁎ with ‖ ϕ ‖ = ‖ x ⁎ ‖ ≡ r such that | 〈 ϕ , f ( x ) 〉 − 〈 x ⁎ , x 〉 | ⩽ 4 e r , for all x ∈ X . Making use of it, we prove that if Y is reflexive and if E ⊂ Y [the annihilator of the subspace F ⊂ Y ⁎ consisting of all functionals bounded on co ¯ ( f ( X ) , − f ( X ) ) ] is α -complemented in Y , then there is a bounded linear operator T : Y → X with ‖ T ‖ ⩽ α such that ‖ T f ( x ) − x ‖ ⩽ 4 e , for all x ∈ X . If, in addition, Y is Gateaux smooth, strictly convex and admitting the Kadec–Klee property (in particular, locally uniformly convex), then we have the following sharp estimate ‖ T f ( x ) − x ‖ ⩽ 2 e , for all x ∈ X .

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b7642bddf752567fd46a3ac644c559251f6372d4