Assume that $X$ and $Y$ are real Banach spaces with the same finite dimension. In this paper we show that if a standard coarse isometry $f:X\rightarrow Y$ satisfies an integral convergence condition or weak stability on a basis, then there exists a surjective linear isometry $U:X\rightarrow Y$ such that $\|f(x)-Ux\|=o(\|x\|)$ as $\|x\|\rightarrow\infty$. This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity. As a consequence, we also obtain a stability result for $\varepsilon$-isometries which was established by Dilworth.
Yuqi SUN
,
Wen ZHANG
. COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES[J]. Acta mathematica scientia, Series B, 2021
, 41(5)
: 1493
-1502
.
DOI: 10.1007/s10473-021-0506-5
[1] Lindenstrauss J, Szankowski A. Non linear perturbations of isometries. Astéisque, 1985, 131:357-371
[2] Benyamini Y, Lindenstrauss J. Geometric Nonlinear Functional Analysis I. Amer Math Soc Colloquium Publications, Vol 48. Providence, RI:Amer Math Soc, 2000
[3] Dolinar G. Generalized stability of isometries. J Math Anal Appl, 2000, 242(1):39-56
[4] Cheng L, Fang Q, Luo S, Sun L. On non-surjective coarse isometries between Banach spaces. Quaestiones Mathematicae, 2019, 42(3):347-362
[5] Bourgain R D. Approximate isometries on finite-dimensional Banach spaces. Trans Amer Math Soc, 1975, 207:309-328
[6] Dilworth S J. Approximate isometries on finite-dimensional normed spaces. Bull London Math Soc, 1999, 31:471-476
[7] Vestfrid I A. Almost surjective ε-isometries of Banach spaces. Colloq Math, 2004, 100:17-22
[8] Cheng L, Dong Y, Zhang W. On stability of non-surjective ε-isometries of Banach spaces. J Funct Anal, 2013, 264(3):713-734
[9] Cheng L, Dai D, Dong Y, Zhou Y. Universal stability of Banach spaces of ε-isometries. Studia Math, 2014, 221(2):141-149
[10] Cheng L, Zhou Y. On preturbed metric-preserved mappings and their stability characterizations. J Funct Anal, 2014, 266(8):4995-5015
[11] Dai D, Dong Y. Stablility of Banach spaces via nonlinear ε-isometries. J Math Anal Appl, 2014, 414:996-1005
[12] Vestfrid I A. Stability of almost surjective ε-isometries of Banach spaces. J Funct Anal, 2015, 269(7):2165-2170
[13] Zhou Y, Zhang Z, Liu C. Stability of ε-isometric embeddings into Banach spaces of continuous functions. J Math Anal Appl, 2017, 453:805-820
[14] Cheng L, Cheng Q, Tu K, Zhang J. A universal theorem for stability of ε-isometries of Banach spaces. J Funct Anal, 2015, 269(1):199-214
[15] Cheng L, Dong Y. Corrigendum to "A universal theorem for stability of ε-isometries of Banach spaces"[J Funct Anal, 2015, 269(1):199-214]. J Funct Anal, 2020, 279:108518
[16] Mazur S, Ulam S. Sur les transformations isométriques d'espaces vectoriels normés. C R Acad Sci Paris, 1932, 194:946-948
