In this article, we discuss the stability of ε-isometries for L∞,λ-spaces. Indeed, we first study the relationship among separably injectivity, injectivity, cardinality injectivity and universally left stability of L∞,λ-spaces as well as we show that the second duals of universally left-stable spaces are injective, which gives a partial answer to a question of Bao-Cheng-Cheng-Dai, and then we prove a sharpen quantitative and generalized Sobczyk theorem which gives examples of nonseparable L∞-spaces X (but not injective) such that the couple (X, Y) is stable for every separable space Y. This gives a new positive answer to Qian's problem.
Duanxu DAI
. STABILITY OF ε-ISOMETRIES ON L∞-SPACES[J]. Acta mathematica scientia, Series B, 2019
, 39(6)
: 1733
-1742
.
DOI: 10.1007/s10473-019-0619-2
[1] Albiac F, Kalton N J. Topics in Banach Space Theory. Graduate Texts in Mathematics 233. New York:Springer, 2006
[2] Argyros S A, Haydon R G. A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem. Acta Math, 2011, 206(1):1-54
[3] Avilés A, Sánchez F C, Castillo Jesús M F, González M, Moreno Y. On separably injective Banach Spaces. Adv Math, 2013, 234:192-216
[4] Bao L, Cheng L, Cheng Q, Dai D. On universally left-stability of ε-isometry. Acta Math Sin, Engl Ser, 2013, 29(11):2037-2046
[5] Benyamini Y, Lindenstrauss J. Geometric Nonlinear Functional Analysis I. Amer Math Soc Colloquium Publications, Vol 48. Providence, RI:Amer Math Soc, 2000
[6] Bourgain J, Delbaen F. A class of special L∞-space. Acta Math, 1980, 145:155-176
[7] Cheng L, Dong Y, Zhang W. On stability of nonlinear non-surjective ε-isometries of Banach spaces. J Funct Anal, 2013, 264:713-734
[8] Cheng L, Dai D, Dong Y, et al. Universal stability of Banach spaces for ε-isometries. Studia Math, 2014, 221:141-149
[9] Cheng L, Cheng Q, Tu K, Zhang J. A universal theorem for stability of ε-isometries of Banach spaces. J Funct Anal, 2015, 269:199-214
[10] Dai D, Dong Y. On stability of Banach spaces via nonlinear ε-isometries. J Math Anal Appl, 2014, 414:996-1005
[11] Dai D, Zheng B. Stability of a pair of Banach spaces for ε-isometries. Acta Math Sci, 2019, 39B(4):1163-1172
[12] Dilworth S J. Approximate isometries on finite-dimensional normed spaces. Bull London Math Soc, 1999, 31:471-476
[13] Dutrieux Y, Lancien G. Isometric embeddings of compact spaces into Banach spaces. J Funct Anal, 2008, 255:494-501
[14] Fabian M, Habala P, Hájek P, Montesinos V, Zizler V. Banach Space Theory. The Basis for Linear and Nonlinear Analysis. New York:Springer, 2011
[15] Figiel T. On non linear isometric embeddings of normed linear spaces. Bull Acad Polon Sci Math Astro Phys, 1968, 16:185-188
[16] Gevirtz J. Stability of isometries on Banach spaces. Proc Amer Math Soc, 1983, 89:633-636
[17] Godefroy G, Kalton N J. Lipschitz-free Banach spaces. Studia Math, 2003, 159:121-141
[18] Gruber P M. Stability of isometries. Trans Amer Math Soc, 1978, 245:263-277
[19] Hyers D H, Ulam S M. On approximate isometries. Bull Amer Math Soc, 1945, 51:288-292
[20] Johnson W B. Communication. mathoverflow, 2012. http://mathoverflow.net/questions/72750/densitycharacter-and-cardinality
[21] Mazur S, Ulam S. Sur les transformations isométriques d'espaces vectoriels normés. C R Acad Sci Paris, 1932, 194:946-948
[22] Omladič M, Šemrl P. On non linear perturbations of isometries. Math Ann, 1995, 303:617-628
[23] Qian S. ε-Isometric embeddings. Proc Amer Math Soc, 1995, 123:1797-1803
[24] Šemrl P, Väisälä J. Nonsurjective nearisometries of Ban ach spaces. J Funct Anal, 2003, 198:268-278
[25] Sobczyk A. Projection of the space (m) on its subspace c0. Bull Amer Math Soc, 1941, 47:938-947
[26] Tabor J. Stability of surjectivity. J Approx Theory, 2000, 105:166-175
[27] Wolfe J. Injective Banach spaces of type C(T). Israel J Math, 1974, 18:133-140
[28] Zippin M. The separable extension problem. Israel J Math, 1977, 26:372-387
