C_0半群的范数函数与临界点
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摘要
本文主要考虑Banach空间上C_0半群{T(t)}_(t≥0)的范数函数t→∥T(t)∥的刻划问题,以及模连续(紧,可微)C_0半群在其模连续(紧,可微)区间的左端点处是否仍然保持模连续(紧,可微)性质的问题。
本文分为四章,第一章中介绍了后续各章的研究背景和主要结果。
第二章中简要叙述了一些基础知识,这些知识涉及泛函分析、实分析和矩阵论,并证明了若干引理。
第三章中研究C_0半群范数函数的刻划问题。首先通过简单的观察和论证给出四条基本性质,它们是一个函数成为某个C_0半群范数函数的必要条件。在§3.2中证明了有限维空间中C_0半群范数的一个增长阶估计,并由此构造一个函数,它满足四条基本性质而不满足增长阶估计,从而不能成为任何有限维Banach空间C_0半群的范数函数。在§3.3中通过在四条基本性质的基础上附加少量假设,证明了满足这些性质和假设即可保证一个函数成为某个无穷维空间上C_0半群的范数函数。
第四章中,我们通过举例说明,一个模连续(紧,可微)半群在其临界点,即模连续(紧,可微)区间的左端点处,既有可能继续保持模连续(紧,可微)性质,也有可能不再保持这些性质,不能就此作出一般性的论断。
In this paper, we consider the characterization of the norm function, t→∥T(t)∥, of C_0-semigroups on Banach spaces, and the problem whether a norm-continuous (compact, differentiable) semigroup remains norm-continuous (com-pact, differentiable) at the left endpoint of its interval of norm-continuity (com-pactness, differentiability).
The thesis consists of four parts. Part one states the background of subse-quent studies and the main results.
Part two contains some basic knowledge involving functional analysis, realanalysis and matrix theory, and the proofs of several preliminary propositions.
In part three, we studies the characterization of the norm function of C_0-semigroups. First of all, by easy reasoning a set of four basic properties is es-tablished as a necessary condition for a function to be the norm function of aC_0-semigroup. In§3.2, a growth estimate of the norm function of C_0-semigroupson ffnite-dimensional spaces is reached, which enable us to construct a functionthat satisffes the four basic properties but fails to meet the growth estimate andhence cannot be the the norm function of any C_0-semigroup on ffnite-dimensionalspace. In§3.3 we make a few assumptions in addition to the four basic proper-ties and proves their suffciency for a function to be the the norm function of aC_0-semigroups on an inffnite-dimensional Banach space.
In the last part, we show by examples that a norm-continuous (compact,differentiable) semigroup may or may not remain norm-continuous (compact, d-ifferentiable) at its critical point, i.e. the left endpoint of its interval of norm- continuity (compactness, differentiability), depending on the concrete situation,and a general conclusion cannot be drawn.
本文分为四章,第一章中介绍了后续各章的研究背景和主要结果。
第二章中简要叙述了一些基础知识,这些知识涉及泛函分析、实分析和矩阵论,并证明了若干引理。
第三章中研究C_0半群范数函数的刻划问题。首先通过简单的观察和论证给出四条基本性质,它们是一个函数成为某个C_0半群范数函数的必要条件。在§3.2中证明了有限维空间中C_0半群范数的一个增长阶估计,并由此构造一个函数,它满足四条基本性质而不满足增长阶估计,从而不能成为任何有限维Banach空间C_0半群的范数函数。在§3.3中通过在四条基本性质的基础上附加少量假设,证明了满足这些性质和假设即可保证一个函数成为某个无穷维空间上C_0半群的范数函数。
第四章中,我们通过举例说明,一个模连续(紧,可微)半群在其临界点,即模连续(紧,可微)区间的左端点处,既有可能继续保持模连续(紧,可微)性质,也有可能不再保持这些性质,不能就此作出一般性的论断。
In this paper, we consider the characterization of the norm function, t→∥T(t)∥, of C_0-semigroups on Banach spaces, and the problem whether a norm-continuous (compact, differentiable) semigroup remains norm-continuous (com-pact, differentiable) at the left endpoint of its interval of norm-continuity (com-pactness, differentiability).
The thesis consists of four parts. Part one states the background of subse-quent studies and the main results.
Part two contains some basic knowledge involving functional analysis, realanalysis and matrix theory, and the proofs of several preliminary propositions.
In part three, we studies the characterization of the norm function of C_0-semigroups. First of all, by easy reasoning a set of four basic properties is es-tablished as a necessary condition for a function to be the norm function of aC_0-semigroup. In§3.2, a growth estimate of the norm function of C_0-semigroupson ffnite-dimensional spaces is reached, which enable us to construct a functionthat satisffes the four basic properties but fails to meet the growth estimate andhence cannot be the the norm function of any C_0-semigroup on ffnite-dimensionalspace. In§3.3 we make a few assumptions in addition to the four basic proper-ties and proves their suffciency for a function to be the the norm function of aC_0-semigroups on an inffnite-dimensional Banach space.
In the last part, we show by examples that a norm-continuous (compact,differentiable) semigroup may or may not remain norm-continuous (compact, d-ifferentiable) at its critical point, i.e. the left endpoint of its interval of norm- continuity (compactness, differentiability), depending on the concrete situation,and a general conclusion cannot be drawn.
引文
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[6] C. J. K. Batty, On a perturbation theorem of Kaiser and Weis, Semigroup Forum 70 (2005), no. 3,471-474.
[7] C. J. K. Batty, M. D. Blake, S. Srivastava, A non-analytic growth bound for Laplace transformsand semigroups of operators, Integral Equations Operator Theory 45 (2003), no. 2, 125-154.
[8] C. J. K. Batty, T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces.J. Evol. Equ. 8 (2008), no. 4, 765–780.
[9] C. J. K. Batty, J. Liang, T. J. Xiao, On the spectral and growth bound of semigroups associatedwith hyperbolic equations, Adv. Math. 191 (2005), 1–10.
[10] C. J. K. Batty, S. Krol, Perturbations of generators of C0C0-semigroups and resolvent decay, J.Math. Anal. Appl. 367 (2010), no. 2, 434-443.
[11] C. J. K. Batty, Y. Tomilov, Quasi-hyperbolic semigroups, J. Funct. Anal. 258 (2010), no. 11,3855-3878.
[12] M. Benchohra, L. G’orniewicz, S. K. Ntouyas, A. Ouahab, Controllability results for nondenselydeffned semilinear functional differential equations. Z. Anal. Anwend. 25 (2006), no. 3, 311–325.
[13] J. Bierkens, O. van Gaans, S V. Lunel, Existence of an invariant measure for stochastic evolutionsdriven by an eventually compact semigroup, J. Evol. Equ. 9 (2009), no. 4, 771–786.
[14]陈公宁,矩阵理论与应用(第2版),北京:科学出版社, 2007.
[15] A. C’wiszewski, Degree theory for perturbations ofm-accretive operators generating compact semi-groups with constraints, J. Evol. Equ. 7 (2007), no. 1, 1–33.
[16] V. Casarino, K. L. Engel, R. Nagel, G. Nickel, A semigroup approach to boundary feedback systems,Integral Equations Operator Theory 47 (2003), no. 3, 289-306.
[17] R. F. Curtain, H. Zwart, An introduction to inffnite-dimensional linear systems theory, Texts inApplied Mathematics, 21. Springer-Verlag, New York, 1995.
[18] J. P. Dauer, N. I. Mahmudov, Exact null controllability of semilinear integrodifferential systems inHilbert spaces, J. Math. Anal. Appl. 299 (2004), no. 2, 322–332.
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[22] R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notesin Math., vol. 1570, Springer-Verlag, Berlin, 1994.
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