STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

doi:10.1007/s10473-019-0418-9

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (4): 1163-1172.doi: 10.1007/s10473-019-0418-9

STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

戴端旭¹, 郑本拓²

1. College of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China;
2. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

收稿日期:2018-03-22 修回日期:2018-09-18 出版日期:2019-08-25 发布日期:2019-09-12
通讯作者: Duanxu DAI E-mail:dduanxu@163.com
作者简介:Bentuo ZHENG,E-mail:bzheng@memphis.edu
基金资助:
Duanxu Dai is supported in part by NSFC (11601264, 11471270 and 11471271), the Fundamental Research Funds for the Central Universities (20720160037), the Outstanding Youth Scientific Research Personnel Training Program of Fujian Province, the High level Talents Innovation and Entrepreneurship Project of Quanzhou City (2017Z032), the Research Foundation of Quanzhou Normal University (2016YYKJ12), and the Natural Science Foundation of Fujian Province of China (2019J05103). Bentuo Zheng is supported in part by NSFC (11628102).

STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

Duanxu DAI¹, Bentuo ZHENG²

1. College of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China;
2. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

Received:2018-03-22 Revised:2018-09-18 Online:2019-08-25 Published:2019-09-12
Supported by:
Duanxu Dai is supported in part by NSFC (11601264, 11471270 and 11471271), the Fundamental Research Funds for the Central Universities (20720160037), the Outstanding Youth Scientific Research Personnel Training Program of Fujian Province, the High level Talents Innovation and Entrepreneurship Project of Quanzhou City (2017Z032), the Research Foundation of Quanzhou Normal University (2016YYKJ12), and the Natural Science Foundation of Fujian Province of China (2019J05103). Bentuo Zheng is supported in part by NSFC (11628102).

摘要/Abstract

摘要： A pair of Banach spaces (X, Y) is said to be stable if for every ε-isometry f:X → Y, there exist γ > 0 and a bounded linear operator T:L(f) → X with ||T|| ≤ α such that ||Tf(x)-x|| ≤ γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.

关键词: Stability, ε-isometry, Figiel theorem, Banach space

Abstract: A pair of Banach spaces (X, Y) is said to be stable if for every ε-isometry f:X → Y, there exist γ > 0 and a bounded linear operator T:L(f) → X with ||T|| ≤ α such that ||Tf(x)-x|| ≤ γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.

Key words: Stability, ε-isometry, Figiel theorem, Banach space

中图分类号:

46B04

戴端旭, 郑本拓. STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES[J]. 数学物理学报(英文版), 2019, 39(4): 1163-1172.

Duanxu DAI, Bentuo ZHENG. STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES[J]. Acta mathematica scientia,Series B, 2019, 39(4): 1163-1172.

参考文献

[1] Argyros S A, Castillo J M F, Granero A S, et al. Complementation and embeddings of c0(I) in Banach spaces. Proc Lond Math Soc, 2002, 85:742-772
[2] Avilés A, Sánchez F C, Castillo Jesús M F, et al. On separably injective Banach Spaces. Advances in Mathematics, 2013, 234:192-216
[3] 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
[4] Benyamini Y. An M-space which is not isomorphic to a C(K)-space. Israel J Math, 1977, 28:98-104
[5] Benyamini Y, Lindenstrauss J. Geometric nonlinear functional analysis. Vol 1//American Mathematical Society Colloquium Publications. Vol 48. Providence, RI:American Mathematical Society, 2000
[6] 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
[7] Cheng L, Dai D, Dong Y, et al. Universal stability of Banach spaces for ε-isometries. Studia Math, 2014, 221:141-149
[8] Cheng L, Dong Y, Zhang W. On stability of nonlinear non-surjective ε-isometries of Banach spaces. J Funct Anal, 2013, 264:713-734
[9] Dai D, Dong Y. On stability of Banach spaces via nonlinear ε-isometries. J Math Anal Appl, 2014, 414:996-1005
[10] Dilworth S J. Approximate isometries on finite-dimensional normed spaces. Bull London Math Soc, 1999, 31:471-476
[11] Figiel T. On non linear isometric embeddings of normed linear spaces. Bull Acad Polon Sci Math Astro Phys, 1968, 16:185-188
[12] Holmes R B. Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics. Vol 24. New York:Springer, 1975
[13] Hyers D H, Ulam S M. On approximate isometries. Bull Amer Math Soc, 1945, 51:288-292
[14] Johnson W B, Lindenstrauss J. Some remarks on weakly compactly generated Banach spaces. Israel J Math, 1974, 17:219-230
[15] Johnson W B, Oikhberg T. Separable lifting property and extensions of local reflexivity. Illinois J Math, 2001, 45:123-137
[16] Kalton N. Extending Lipschitz maps into C(K)-spaces. Israel J Math, 2007, 162:275-315
[17] Lindenstrauss J. On the extension of operators with range in a C(K) space. Proc Amer Math Soc, 1964, 15:218-225
[18] Mazur S, Ulam S. Sur les transformations isométriques d'espaces vectoriels normés. C R Acad Sci Paris, 1932, 194:946-948
[19] Omladič M, Šemrl P. On non linear perturbations of isometries. Math Ann, 1995, 303:617-628
[20] Phelps R R. Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics 1364. Springer, 1989
[21] Qian S. ε-Isometric embeddings. Proc Amer Math Soc, 1995, 123:1797-1803
[22] Šemrl P, Väisälä J. Nonsurjective nearisometries of Banach spaces. J Funct Anal, 2003, 198:268-278
[23] Tabor J. Stability of surjectivity. J Approx Theory, 2000, 105:166-175

STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Temur JANGVELADZE, Zurab KIGURADZE, Mikheil GAGOSHIDZE. ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM[J]. 数学物理学报(英文版), 2019, 39(4): 971-988.
[2]	Prondanai KASKASEM, Chakkrid KILN-EAM. HYPERSTABILITY OF THE GENERALIZED CAUCHY-JENSEN FUNCTIONAL EQUATION IN ULTRAMETRIC SPACES[J]. 数学物理学报(英文版), 2019, 39(4): 1017-1032.
[3]	江一鸣, 王苏鑫, 王兴春. ASYMPTOTICS OF THE SOLUTIONS TO STOCHASTIC WAVE EQUATIONS DRIVEN BY A NON-GAUSSIAN LEVY PROCESS[J]. 数学物理学报(英文版), 2019, 39(3): 731-746.
[4]	Iz-iddine EL-FASSI, Hamid KHODAEI, Themistocles M. RASSIAS. APPROXIMATE SOLUTION OF A p-th ROOT FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN (2, β)-BANACH SPACES[J]. 数学物理学报(英文版), 2019, 39(2): 369-381.
[5]	Muaadh ALMAHALEBI, Abdellatif CHAHBI. APPROXIMATE SOLUTION OF P-RADICAL FUNCTIONAL EQUATION IN 2-BANACH SPACES[J]. 数学物理学报(英文版), 2019, 39(2): 551-566.
[6]	刘春根, 张晓飞. STABILITY OF SUBHARMONIC SOLUTIONS OF FIRST-ORDER HAMILTONIAN SYSTEMS WITH ANISOTROPIC GROWTH[J]. 数学物理学报(英文版), 2019, 39(1): 111-126.
[7]	潘洁, 方莉, 郭真华. STABILITY OF BOUNDARY LAYER TO AN OUTFLOW PROBLEM FOR A COMPRESSIBLE NON-NEWTONIAN FLUID IN THE HALF SPACE[J]. 数学物理学报(英文版), 2019, 39(1): 259-283.
[8]	张海湘, 杨雪花, 徐大. ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL[J]. 数学物理学报(英文版), 2018, 38(6): 1689-1711.
[9]	刘丽雅, 蒋达清, Tasawar HAYAT, Bashir AHMAD. DYNAMICS OF A HEPATITIS B MODEL WITH SATURATED INCIDENCE[J]. 数学物理学报(英文版), 2018, 38(6): 1731-1750.
[10]	于林, 王汝慧, 赵守江. CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES[J]. 数学物理学报(英文版), 2018, 38(6): 1779-1788.
[11]	薛焕斌, 张继业. ROBUST EXPONENTIAL STABILITY OF SWITCHED DELAY INTERCONNECTED SYSTEMS UNDER ARBITRARY SWITCHING[J]. 数学物理学报(英文版), 2018, 38(6): 1921-1938.
[12]	陈贵强, Matthew RIGBY. STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW[J]. 数学物理学报(英文版), 2018, 38(5): 1485-1514.
[13]	Jasbir Singh MANHAS, Ruhan ZHAO. PRODUCTS OF WEIGHTED COMPOSITION AND DIFFERENTIATION OPERATORS INTO WEIGHTED ZYGMUND AND BLOCH SPACES[J]. 数学物理学报(英文版), 2018, 38(4): 1105-1120.
[14]	周永辉, 杨赟瑞, 刘克盼. STABILITY OF TRAVELING WAVES IN A POPULATION DYNAMIC MODEL WITH DELAY AND QUIESCENT STAGE[J]. 数学物理学报(英文版), 2018, 38(3): 1001-1024.
[15]	唐少君, 张澜. NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION[J]. 数学物理学报(英文版), 2018, 38(3): 973-1000.