F*空间上点态半有界和非无界线性算子族的共鸣定理

数学物理学报 ›› 2014, Vol. 34 ›› Issue (5): 1071-1082.

F*空间上点态半有界和非无界线性算子族的共鸣定理

宋明亮

江苏第二师范学院数学与信息技术学院南京 |210013

收稿日期:2013-10-09 修回日期:2014-08-04 出版日期:2014-10-25 发布日期:2014-10-25
基金资助:
国家自然科学基金(10671094)、江苏省高校“青蓝工程”基金(2010QLP27)和江苏省高校自然科学基金(13KJB110004)资助.

Resonance Theorems for Families of Pointwise Semi-bounded and Non-unbounded Linear Operators on F* Spaces

SONG Ming-Liang

Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing 210013

Received:2013-10-09 Revised:2014-08-04 Online:2014-10-25 Published:2014-10-25
Supported by:
国家自然科学基金(10671094)、江苏省高校“青蓝工程”基金(2010QLP27)和江苏省高校自然科学基金(13KJB110004)资助.

摘要/Abstract

摘要：

证明每个F*空间(即满足第一可数公理的Hausdorff拓扑向量空间)可借助于它的“标准生成伪范数族”来表征. 利用标准生成伪范数族P, 在F*空间中引入P -有界集、P -半有界集和P -无界集的概念, 建立点态半有界和非无界线性算子族的共鸣定理. 作为其推论, 得到了Menger概率赋范空间中点态概率半有界和非概率无界线性算子族的共鸣定理, 改进并推广了某些已有的结果.

关键词: F*空间, P -有界集, P -半有界集, P -无界集, 共鸣定理

Abstract:

We prove that every F* space (i.e., Hausdorff topological vector space satisfying the first countable axiom) can be characterized by meams of its “standard generating family of pseudo-norms”. By using the standard generating family of pseudo-norms P, the concepts of P-bounded set, P-semi-bounded set and P-unbounded set in F* space are introduced. The resonance theorems for families of pointwise semi-bounded and pointwise non-unbounded linear operators on F* spaces are established. As their applications, the resonance theorems for families of pointwise probabilistically semi-bounded and pointwise probabilistically non-unbounded linear operators in Menger probabilistic normed spaces are obtained.

Key words: F* space, P-bounded set, P-semi-bounded set, P-unbounde set, Resonance theorem

中图分类号:

46A16

宋明亮. F*空间上点态半有界和非无界线性算子族的共鸣定理[J]. 数学物理学报, 2014, 34(5): 1071-1082.

SONG Ming-Liang. Resonance Theorems for Families of Pointwise Semi-bounded and Non-unbounded Linear Operators on F* Spaces[J]. Acta mathematica scientia,Series A, 2014, 34(5): 1071-1082.

参考文献

[1] Ding G G. The nesonance theorem for a family of convex functionals. Acta Mathematica Sinica,1981, 24(7): 851--856

[2] Zhu J S, Li H. On resonance theorems. Acta Mathematica Sinica, 1988, 31(2): 192--200. (in Chinese)

[3] Xuan H N. A general form for the uniform bounded principle. Acta Mathematica Sinica, 1991, 34(1): 132--137. (in Chinese)

[4] Huang J, Wang L P. On sufficient conditions for the resonance theorem of convex functionals. Acta Mathematica Sinica, 1993, 36(7): 775--777. (in Chinese)

[5] Roth W. A uniform boundedness theorem for locally convex cones. Proc Amer Math Soc, 1998, 125(8): 1973--1982

[6] Xiao J Z, Zhu X H. Probabilistic norm of operators and resonance theorems. Applied Mathematics and Mechanics, 1999, 20(8): 781--788

[7] Nygaard O. A strong boundedness principle in Banach spaces. Proc Amer Math Soc, 2000, 129(3): 861--863

[8] Liu Y P. Resonance theorem in quasi-normed space. Acta Scientiarum Naturalium Universitatis Nankaiensis. 2003, 36(1): 22--27. (in Chinese)

[9] Xiao J Z, Zhu X H. The uniform boundedness principles for fuzzy normed spaces. The Joumal of Fuzzy Mathematies, 2004, 12(3): 543--560

[10] Qiu J H. Resonance theorems for families of quasi-homogeneous operators. Chinese Ann Math, 2004, 25A(3): 389--396. (in Chinese)

[11] Qiu J H. Resonance theorems for families og quasi-homogeneous operators taking values in topological vector spaces. J Math Res Exposition, 2005, 25(1): 134--138. (in Chinese)

[12] Song M L, Fang J X. Resonance theorems for family of quasi-homogeneous operators in fuzzy normed linear spaces.
Fuzzy Sets and Systems, 2008, 159(7): 708--719

[13] Shou J L, Wang M S. Introduction to Analytic Topology. Xian: Xian Jiaotong University Press, 1988. (in Chinese)

[14] Liu P D. The Basis of Topological Linear Spaces. Wuhan: Wuhan University Press, 2002. (in Chinese)

[15] Ding G G. Selected Topics of Topological Linear Spaces. Nanning: Guangxi Education Publishing House, 1987. (in Chinese)

[16] Schweizer B, Sklar A. Probabilistic Metric Space. New York, Amsterdam, Oxford: North-Holland, 1983

[17] Chang S S, Cho Y J, Kang S M. Probabilistic Metric Spaces and Nonlinear Operator Theory. Chengdu: Sichuan University Press, 1994

[18] Fang J X. On probabilistic norm of a linear operator and space of operators. Applied Mathematics and Mechanics, 1999, 20(10): 1660--1666

[19] Guo T X, Lin X, Gong H Y. Uniformly bounded principle on PN spaces. J Xian Jiaotong University, 1990, 24(1): 65--70. (in Chinese)

[20] Fang J X. On probabilistic normed spaces and the Condition (PN-5). J Nanjing Normal University (Natural Science),
1998, 21(1): 1--10. (in Chinese)