One-parameter Semigroups of Positive Operators
One-parameter Semigroups of Positive Operators
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DOI：
10.1007/bfb0074922
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发表时间：
1986-04
期刊：
影响因子：
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通讯作者：
W. Arendt;A. Grabosch;G. Greiner;Ulrich Moustakas;R. Nagel;U. Schlotterbeck;U. Groh;H. Lotz
中科院分区：
文献类型：
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作者：
W. Arendt;A. Grabosch;G. Greiner;Ulrich Moustakas;R. Nagel;U. Schlotterbeck;U. Groh;H. Lotz
早在1948年第一版的《半群与泛函分析》中，E.希勒表示需要”发展一个适当的理论转换半群运作在偏序空间”（1。（c）前言）。与此同时，单参数正线性算子半群理论也在不断发展。受概率论和偏微分方程问题的启发，Feller（1952）和RS菲利普斯（1962）通过刻画特殊正半群的生成元奠定了第一个基石。在60年代和70年代，有序Banach空间上的正算子理论被系统地建立起来，并在HH Schaefer（1974）和AC Zaanen（1983）的专著中有很好的记载。但在这个过程中，与应用，特别是与初始值问题的原始联系有时被掩盖了。只是在最近几年出现了一个充分的和最新的理论，主要是基于技术开发的积极运营商，从而重组功能分析理论的调查柯西问题有正解，每个积极的初始值。即使这一发展，特别是关于具体的应用演化方程在运输理论，数学生物学和物理学是远远没有完成，目前的卷是第一次尝试塑造众多的可用结果成一个连贯的理论单参数半群的正线性算子有序Banach spaces.The书是组织如下。我们集中我们的注意力在半群理论的三个主题：特征，谱理论和渐近行为。通过刻画，我们理解了通过生成元描述半群的特殊性质（如正性）的问题。我们所说的谱理论是指对发生器的谱的研究.渐近性态是指在给定半群下初值的轨道和稳定性等现象。
As early as 1948 in the first edition of his fundamental treatise on Semigroups and Functional Analysis, E. Hille expressed the need for" developing an adequate theory of transformation semigroups operating in partially ordered spaces"(1. c,, Foreword). In the meantime the theory of one-parameter semigroups of positive linear operators has grown continuously. Motivated by problems in probability theory and partial differential equations W. Feller (1952) and RS Phillips (1962) laid the first cornerstones by characterizing the generators of special positive semigroups. In the 60's and 70's the theory of positive operators on ordered Banach spaces was built systematically and is well documented in the monographs of HH Schaefer (1974) and AC Zaanen (1983). But in this process the original ties with the applications and, in particular, with initial value problems were at times obscured. Only in recent years an adequate and up-to-date theory emerged, largely based on the techniques developed for positive operators and thus recombining the functional analytic theory with the investigation of Cauchy problems having positive solutions to each positive initial value. Even though this development-in particular with respect to applications to concrete evolution equations in transport theory, mathematical biology, and physics is far from being complete, the present volume is a first attempt to shape the multitude of available results into a coherent theory of one-parameter semigroups of positive linear operators on ordered Banach spaces.The book is organized as follows. We concentrate our attention on three subjects of semigroup theory: characterization, spectral theory and asymptotic behavior. By characterization, we understand the problem of describing special properties of a semigroup, such as positivity, through the generator. By spectral theory we mean the investigation of the spectrum of a generator. Asymptotic behavior refers to the orbits of the initial values under a given semigroup and phenomena such as stability.