【作者】 林羽；

【导师】 肖体俊； 梁进；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2011， 博士

【摘要】 本文主要考虑Banach空间上C₀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）C₀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C₀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C₀半群范数函数的必要条件。在§3.2中证明了有限维空间中C₀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C₀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C₀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。更多还原

【Abstract】 In this paper, we consider the characterization of the norm function, t→∥T（t）∥, of C₀-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 C₀-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 aC₀-semigroup. In§3.2, a growth estimate of the norm function of C₀-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 C₀-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 aC₀-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.更多还原