On non-surjective coarse isometries between Banach spaces

Abstract

Assume that X; Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f : X → Y is a standard coarse isometry. In this paper, we show that if

∫ ∞      ε_f(S) ¹/_P             

        __________ ds < ∞,

    1      S¹ + ¹/_p

then there is a linear isometry U : X → Y so that

‖ f (x) - U x ‖ = o (‖ x ‖), as ‖ x ‖ → ∞,

where ε_f : ℝ⁺ → ℝ⁺ is defined by

ε_f(t) = sup {l ‖ f (x) - f (y) ‖ - ‖ l : x,y ∈ X ‖ x - y ‖ ≤ t }.

Representation properties of coarse isometries in free ultrafilter limits on ℕ are also discussed.

Mathematics Subject Classification (2010): Primary 46B04, 46B20, 47A58; Secondary 46A20.

Keywords: Coarse isometry, stability, uniform convexity, Banach space