正则预解算子族的谱与连续性
|
摘要
本文主要研究了C-正则预解算子族的几个重要性质,所做工作是在预解算子族相关性质的研究基础上展开,所得结论推广了预解算子族的结论,包含了C_0-半群、C-正则半群、余弦算子族和C-正则余弦算子族等的结果.本文共分四部分.
第一章,我们简要地介绍了C-正则预解算子族的历史与发展情况.
第二章,我们首先介绍了C-正则预解算子族的一些重要性质;然后我们给出了C-正则预解算子族的两种无穷小生成元定义,并讨论了它们之间的联系;在第四节中,我们给出了C-正则预解算子族的谱的概念,在此基础上研究了C-正则预解算子族的谱与其生成元的谱之间的关系,得到了C-正则预解算子族的谱映像定理;最后一节讨论了C-正则预解算子族的共轭算子,给出了C-正则预解算子族的共轭算子是C-正则预解算子族的条件.
第三章,我们讨论了C-正则预解算子族的连续性与稳定性.第二节中,我们讨论了不同假设条件下C-正则预解算子族的一致连续性:若R(t)是Banach空间X上指数有界的C-正则预解算子族, R(t)一致连续的充要条件是其生成元A是Banach空间X上的有界算子;若R(t)是Hilbert空间H上指数稳定的C-正则预解算子族,那么R(t)一致连续等价于其预解式H(iμ)沿虚轴衰减到零.本章最后一节讨论了C-正则预解算子族的一致稳定性,给出了C-正则预解算子族一致稳定的充分条件.
第四章,总结全文并提出展望.
In this thesis, some properties of C-regularized resolvent family have been studied. Thiswork bases on the studying of resolvent family, it extents the concept of resolvent family of op-erator, all the conclusions in the thesis have includes the corresponding results of C_0-semigroup,C-semigroup and resolvent family . There’re four chapters in this thesis.
Chapter 1 is preface, it simply introduces the history and development of C-regularizedresolvent family.
In Chapter 2, we first introduce some useful properties of C-regularized resolvent family .Then two definitions of infinitesimal generator are given in Section 3. Based on the spectrum ofC-regularized resolvent family, we study the relationship between spectrum of C-regularizedresolvent family and of its generator, and gain the spectral mapping theorem in section 4. In thelast section , adjoint operator of C-regularized resolvent family is studied.
Chapter 3 is about continuity and stability of C-regularized resolvent family. We discussthe uniform continuity of C-regularized resolvent family on the different condition : if R(t) isexponential bounded C-regularized resolvent family on Banach space X, it’s uniform continuityif and only if its generator is a bounded operator on Banach space X; if R(t) is exponentialstability on Hilbert space H, uniform continuity is equivalent to the decay to zero of a holomorphicoperator family along some imaginary axis. In last section, we give sufficient condition for theuniform stability of C-regularized resolvent family.
In Chapter 4 ,all work discussed above is concluded and further study directions are putforward.
第一章,我们简要地介绍了C-正则预解算子族的历史与发展情况.
第二章,我们首先介绍了C-正则预解算子族的一些重要性质;然后我们给出了C-正则预解算子族的两种无穷小生成元定义,并讨论了它们之间的联系;在第四节中,我们给出了C-正则预解算子族的谱的概念,在此基础上研究了C-正则预解算子族的谱与其生成元的谱之间的关系,得到了C-正则预解算子族的谱映像定理;最后一节讨论了C-正则预解算子族的共轭算子,给出了C-正则预解算子族的共轭算子是C-正则预解算子族的条件.
第三章,我们讨论了C-正则预解算子族的连续性与稳定性.第二节中,我们讨论了不同假设条件下C-正则预解算子族的一致连续性:若R(t)是Banach空间X上指数有界的C-正则预解算子族, R(t)一致连续的充要条件是其生成元A是Banach空间X上的有界算子;若R(t)是Hilbert空间H上指数稳定的C-正则预解算子族,那么R(t)一致连续等价于其预解式H(iμ)沿虚轴衰减到零.本章最后一节讨论了C-正则预解算子族的一致稳定性,给出了C-正则预解算子族一致稳定的充分条件.
第四章,总结全文并提出展望.
In this thesis, some properties of C-regularized resolvent family have been studied. Thiswork bases on the studying of resolvent family, it extents the concept of resolvent family of op-erator, all the conclusions in the thesis have includes the corresponding results of C_0-semigroup,C-semigroup and resolvent family . There’re four chapters in this thesis.
Chapter 1 is preface, it simply introduces the history and development of C-regularizedresolvent family.
In Chapter 2, we first introduce some useful properties of C-regularized resolvent family .Then two definitions of infinitesimal generator are given in Section 3. Based on the spectrum ofC-regularized resolvent family, we study the relationship between spectrum of C-regularizedresolvent family and of its generator, and gain the spectral mapping theorem in section 4. In thelast section , adjoint operator of C-regularized resolvent family is studied.
Chapter 3 is about continuity and stability of C-regularized resolvent family. We discussthe uniform continuity of C-regularized resolvent family on the different condition : if R(t) isexponential bounded C-regularized resolvent family on Banach space X, it’s uniform continuityif and only if its generator is a bounded operator on Banach space X; if R(t) is exponentialstability on Hilbert space H, uniform continuity is equivalent to the decay to zero of a holomorphicoperator family along some imaginary axis. In last section, we give sufficient condition for theuniform stability of C-regularized resolvent family.
In Chapter 4 ,all work discussed above is concluded and further study directions are putforward.
引文
[1] Pazy A. Semigroups of linear operators and applications to partial differential equations[M]. New York:Springer, 1983.
[2]周鸿兴,王连文.线性算子半群理论及应用[M].山东:山东科学技术出版社, 1994.
[3] Da Prato. G. Semigruppi regolazizzabili. Ricerehe Math[J]. 1966, 15: 223-248.
[4] Davies E.B. and Pang M.M.H. . The Cauchy problem and a generalization of the Hille-Yosida theorem.Proc. London Math.Soc[J]. 1987, 55(3): 181-208.
[5] N. Tanaka and I. Miyadera. Exponentially bound C?semigroup and integrated semigroup. Tokyo J.Math[J]. 1989, (1): 99-115.
[6] N. Tanaka. On perturbation theory for exponentially bound C?semigroup.Semigroup Foruum[J]. 1990,41(2): 215-236.
[7] N. Tanaka and N. Okazawa. Local C?semigroup and local integrated semigroups. Proc. LondonMath.Soc.(3)[J]. 1990, 61(1): 63-90.
[8] N. Tanaka and I. Miyadera. C?semigroup and the Abstract Caunchy Problem. Math. Anal. Appl. [J]. 1992,170(1): 196-206.
[9]施德明,杨录山.局部等度连续C?半群.系统科学与数学[J]. 1994, 14(2): 121-129.
[10]姚景齐.指数有界C?半群的共轭半群.系统科学与数学[J]. 1999, 7: 319-322.
[11]施德明,杨录山.指数有界C?半群的Laplace逆变换.郑州大学学报(自然科学版)[J]. 1994, 26(2):1-8.
[12]李淼,黄发伦.正则半群的范数连续性.四川大学学报(自然科学版)[J]. 2001, 38(6): 787-792.
[13]黄廷文,黄发伦. C?半群的解析性及其扰动.四川大学学报(自然科学版)[J]. 1994, 31(3): 287-291.
[14]杜省权,刘清荣. C?半群的谱与其生成元谱之间的关系.纯粹数学与应用数学[J]. 2002, 11: 56-60.
[15] Melnikova I. V. and Filinkov A. I. . Abstract Cauchy problems:three approaches[M]. New York: Chapman￡Hall, CRC, 1999.
[16] C. Greiner and Nagel. On the stability of strong continuous semigroups of positive operators In L2(μ)[J].Ann. Scuola Norm. Sup. PisaSer. 1983, 10(4): 257-262.
[17] W. Arendt and J. Prüss. Vector-valued tauberian theorem and asymptotic behaviour of linear Volterraequations. SIAM J. Math. Anal. [J]. 1992, 23(2): 412-448.
[18] J. Prüss. Evolutionary integral equations and applications[M]. Basel; Boston; Berlin: Monographs inMathematics, 1993, 87, Birkha¨user.
[19] Lizama Carlos. A mean ergodic theoren for resolvent operators. Semigroup Forum[J]. 1993, 47: 227-230.
[20] J. C. Chang and S. -Y Shan. Multiplicative and additive perturbations of resolvent families. Int. Math. [J].200, 2: 841-859.
[21] Lizama Carlos. Uniform Continuity and compactness for resolvent families of operators. Acta ApplicandaeMathematicae[J], 1995, 38: 131-138.
[22] Lizama Carlos. A representation formula for strongly continuous resolvent families. Integral Equationsand Appl. [J], 1997, 9: 278-292.
[23] Lizama Carlos. A Characterization of Uniform Continuity for Volterra equations in Hilbert Spaces. Amer-ican Mathematical Society[J]. 1998, 126: 3581-3587.
[24] Lizama Carlos. Uniform stability of resolvent families. American Mathematical Society[J]. 2003, 132:175-181.
[25] Lizama Carlos. A representation formula for strongly continuous resolvent families. Integral EquationsAppl[J]. 1997, 9(4): 321-327.
[26]郑权,孙应传.预解算子族的一个推广.数学物理学报[J]. 2002, 20(增刊): 723-726.
[27] Miao Li, Quan Zheng and Jizhou Zhang. Regularized Resolvent Families. Taiwanese. Math. [J]. 2007, 11:117-133.
[28] Yu Xin and Chen Liang. Multiplicative perturbations of C?regularized resolvent families. Journal of Zhe-jiang University Science[J]. 2004, 5(5): 528-532.
[29]张寄洲. C?正则预解算子族的遍历性.数学研究与评论[J]. 2005, 25(1): 122-127.
[30]李亚. C?正则预解算子族的左乘积扰动.浙江大学学报(理学版)[J]. 2007, 34(3): 248-251.
[31]张寄洲,尹丹. k-正则预解族的谱映象定理.上海师范大学学报[J]. 2002, 31: 8-12.
[32]张寄洲.正则余弦函数的谱映象定理.湖北大学学报[J]. 1998, 20: 22-26.
[33] Lizama Carlos. A Characterization of Periodic Resolvent Operators. Results Math[J]. 1990, 18: 93-105.
[34] Wang Shengwang and Huang Zhenyou. Strongly continuous integrated C?consin operator functions. Stu-dia Math.[J]. 1997, 126(3): 273-289.
[35]彭爱民,宋晓秋,张祥之. Hilbert空间上的C-半群的稳定性.徐州师范大学学报(自然科学版)[J]. 2004,22(2): 5-8.
[36]杨录山.指数有界C?半群的收敛性.信息工程学院学报[J]. 1993, 11(2): 63-70.
[37] Doetsch. G. . Introduction to the theroy and application of the Laplace transformation[M]. New York:Springer-Verlag, 1974.
[38] Justino Sánchez. On perturbantion of K-regularized resolvent families. Taiw. Math. [J]. 2003, 7: 217-227.
[39] J. Liang and T. Xiao. Norm continuity (for t>0) of propagators of arbitrary order abstract differentialeauations in Hilbert spaces. Math. Anal. Appl.[J]. 1996, 204: 124-137.
[40] Jizhou Zhang. Regularized Cosine Functions and Polynomials of the Groups Generators. Scienitia Iran-ica[J]. 1997, 4(2): 12-18.
[41] Wang Haiyan. The approximation of strongly continuous C?cosine operator functions. Nanjing DaxueXuebao Shuxue Bang Kan[J]. 1995, 11(2): 233-240.
[42] N. Tanaka and I. Miyadera. Some remarks on C?semigroups and intcgrated semigroups. Proc. JapanAcad. Ser. A Math. Sci.[J]. 1987, 63(5): 139-142.
[43] Wang Haiyan. C?cosine operator functions and the second order abstract Cauchy problem. Northeast.Math. [J]. 1995, 11(1): 85-94.
[44] Lizama Carlos. Regularized solutions for volterra equations. Math. Anal. Appl.[J]. 2000, 243: 278-292.
[2]周鸿兴,王连文.线性算子半群理论及应用[M].山东:山东科学技术出版社, 1994.
[3] Da Prato. G. Semigruppi regolazizzabili. Ricerehe Math[J]. 1966, 15: 223-248.
[4] Davies E.B. and Pang M.M.H. . The Cauchy problem and a generalization of the Hille-Yosida theorem.Proc. London Math.Soc[J]. 1987, 55(3): 181-208.
[5] N. Tanaka and I. Miyadera. Exponentially bound C?semigroup and integrated semigroup. Tokyo J.Math[J]. 1989, (1): 99-115.
[6] N. Tanaka. On perturbation theory for exponentially bound C?semigroup.Semigroup Foruum[J]. 1990,41(2): 215-236.
[7] N. Tanaka and N. Okazawa. Local C?semigroup and local integrated semigroups. Proc. LondonMath.Soc.(3)[J]. 1990, 61(1): 63-90.
[8] N. Tanaka and I. Miyadera. C?semigroup and the Abstract Caunchy Problem. Math. Anal. Appl. [J]. 1992,170(1): 196-206.
[9]施德明,杨录山.局部等度连续C?半群.系统科学与数学[J]. 1994, 14(2): 121-129.
[10]姚景齐.指数有界C?半群的共轭半群.系统科学与数学[J]. 1999, 7: 319-322.
[11]施德明,杨录山.指数有界C?半群的Laplace逆变换.郑州大学学报(自然科学版)[J]. 1994, 26(2):1-8.
[12]李淼,黄发伦.正则半群的范数连续性.四川大学学报(自然科学版)[J]. 2001, 38(6): 787-792.
[13]黄廷文,黄发伦. C?半群的解析性及其扰动.四川大学学报(自然科学版)[J]. 1994, 31(3): 287-291.
[14]杜省权,刘清荣. C?半群的谱与其生成元谱之间的关系.纯粹数学与应用数学[J]. 2002, 11: 56-60.
[15] Melnikova I. V. and Filinkov A. I. . Abstract Cauchy problems:three approaches[M]. New York: Chapman￡Hall, CRC, 1999.
[16] C. Greiner and Nagel. On the stability of strong continuous semigroups of positive operators In L2(μ)[J].Ann. Scuola Norm. Sup. PisaSer. 1983, 10(4): 257-262.
[17] W. Arendt and J. Prüss. Vector-valued tauberian theorem and asymptotic behaviour of linear Volterraequations. SIAM J. Math. Anal. [J]. 1992, 23(2): 412-448.
[18] J. Prüss. Evolutionary integral equations and applications[M]. Basel; Boston; Berlin: Monographs inMathematics, 1993, 87, Birkha¨user.
[19] Lizama Carlos. A mean ergodic theoren for resolvent operators. Semigroup Forum[J]. 1993, 47: 227-230.
[20] J. C. Chang and S. -Y Shan. Multiplicative and additive perturbations of resolvent families. Int. Math. [J].200, 2: 841-859.
[21] Lizama Carlos. Uniform Continuity and compactness for resolvent families of operators. Acta ApplicandaeMathematicae[J], 1995, 38: 131-138.
[22] Lizama Carlos. A representation formula for strongly continuous resolvent families. Integral Equationsand Appl. [J], 1997, 9: 278-292.
[23] Lizama Carlos. A Characterization of Uniform Continuity for Volterra equations in Hilbert Spaces. Amer-ican Mathematical Society[J]. 1998, 126: 3581-3587.
[24] Lizama Carlos. Uniform stability of resolvent families. American Mathematical Society[J]. 2003, 132:175-181.
[25] Lizama Carlos. A representation formula for strongly continuous resolvent families. Integral EquationsAppl[J]. 1997, 9(4): 321-327.
[26]郑权,孙应传.预解算子族的一个推广.数学物理学报[J]. 2002, 20(增刊): 723-726.
[27] Miao Li, Quan Zheng and Jizhou Zhang. Regularized Resolvent Families. Taiwanese. Math. [J]. 2007, 11:117-133.
[28] Yu Xin and Chen Liang. Multiplicative perturbations of C?regularized resolvent families. Journal of Zhe-jiang University Science[J]. 2004, 5(5): 528-532.
[29]张寄洲. C?正则预解算子族的遍历性.数学研究与评论[J]. 2005, 25(1): 122-127.
[30]李亚. C?正则预解算子族的左乘积扰动.浙江大学学报(理学版)[J]. 2007, 34(3): 248-251.
[31]张寄洲,尹丹. k-正则预解族的谱映象定理.上海师范大学学报[J]. 2002, 31: 8-12.
[32]张寄洲.正则余弦函数的谱映象定理.湖北大学学报[J]. 1998, 20: 22-26.
[33] Lizama Carlos. A Characterization of Periodic Resolvent Operators. Results Math[J]. 1990, 18: 93-105.
[34] Wang Shengwang and Huang Zhenyou. Strongly continuous integrated C?consin operator functions. Stu-dia Math.[J]. 1997, 126(3): 273-289.
[35]彭爱民,宋晓秋,张祥之. Hilbert空间上的C-半群的稳定性.徐州师范大学学报(自然科学版)[J]. 2004,22(2): 5-8.
[36]杨录山.指数有界C?半群的收敛性.信息工程学院学报[J]. 1993, 11(2): 63-70.
[37] Doetsch. G. . Introduction to the theroy and application of the Laplace transformation[M]. New York:Springer-Verlag, 1974.
[38] Justino Sánchez. On perturbantion of K-regularized resolvent families. Taiw. Math. [J]. 2003, 7: 217-227.
[39] J. Liang and T. Xiao. Norm continuity (for t>0) of propagators of arbitrary order abstract differentialeauations in Hilbert spaces. Math. Anal. Appl.[J]. 1996, 204: 124-137.
[40] Jizhou Zhang. Regularized Cosine Functions and Polynomials of the Groups Generators. Scienitia Iran-ica[J]. 1997, 4(2): 12-18.
[41] Wang Haiyan. The approximation of strongly continuous C?cosine operator functions. Nanjing DaxueXuebao Shuxue Bang Kan[J]. 1995, 11(2): 233-240.
[42] N. Tanaka and I. Miyadera. Some remarks on C?semigroups and intcgrated semigroups. Proc. JapanAcad. Ser. A Math. Sci.[J]. 1987, 63(5): 139-142.
[43] Wang Haiyan. C?cosine operator functions and the second order abstract Cauchy problem. Northeast.Math. [J]. 1995, 11(1): 85-94.
[44] Lizama Carlos. Regularized solutions for volterra equations. Math. Anal. Appl.[J]. 2000, 243: 278-292.