Non-surjective coarse isometrics between Lp spaces

In this paper, we study stability and weak stability of coarse isometrics of Banach spaces. As a result, we show that if a coarse isometry f : L-p(Omega(1), Sigma(1), mu(1)) -> L-p(Omega(2) Sigma(2), mu(2)) (1 < p < infinity) is weakly stable at each point of a Schauder basis, then there is a linear isometry U : L-p(Omega(1), Sigma(1), mu(1)) -> L-p(Omega(2), Sigma(2), mu(2)), where (Omega(j), Sigma(j), mu(j)) (j = 1, 2) are sigma-finite measure spaces. Furthermore, if f is uniformly weakly stable, then parallel to f(x) - Ux parallel to = o(parallel to x parallel to) when parallel to x parallel to -> infinity. As an application, we obtain that parallel to Pf(x) - Ux parallel to = o(parallel to x parallel to) is equivalent to parallel to f(x) - Ux parallel to = o(parallel to x parallel to) as parallel to x parallel to -> infinity, where P : Y -> U(X) is a projection with parallel to P parallel to = 1. (C) 2020 Elsevier Inc. All rights reserved.