Studies of the Structure of Operators on Function Spaces

函数空间算子结构的研究

• 批准号: 10640189
• 负责人: WATANABE Seiji
• 金额: $ 0.7万
• 依托单位: Niigata Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640189/
• 关键词: Function Spaces; Linear Operators; Unbounded Derivations; Weighted Compositi Operators

The theory of function spaces plays an basic role in many branches of pure mathematics, for example, function analysis, differential equations, Fourier analysis, etc. and has become a useful tool in applied mathematics.In this research, we studied the space of diffrentiable functions from the point of view of unbounded derivations.The domain of a unbounded derivation in the space of continuous functions on a compact Hausdoruff space may be regarded as one of generalizations of the space of continuously differentiable functions. We investigated the structure of two important operators (that is, surjective linear isometries and small-bound isomorphisms) on such domain.At first, we decided extreme points of the unit sphere of the conjugate space of the domain equipped with the Cambern norm and used it to prove that surjective linear isometries of the domain are weighted composition operators induced by homeomorphisms of the underlying compact Hausdorff spaces. When the underlying topological spaces satisfy the first countability axiom, this result is extended to the second order derivations.We further studied small-bound isomorphisms on the domain and showed that if there is a small bound-isomorphism, then the underlying topological spaces are homeomorphic.As by-product, we obtained Korovkin type approximation theorems on the space of continuously differentiable functions on the unit interval of the real line.

函数空间理论在函数分析、微分方程、傅立叶分析等纯数学的许多分支中起着基础性的作用，并已成为应用数学中的一个有用工具。本文从无界导子的角度研究了可微函数空间，在紧Hausdoruff空间上的连续函数空间中无界导子的定义域可以是被认为是连续可微函数空间的推广之一。本文研究了这类整环上两个重要算子（满射线性等距和小界同构）的结构，首先确定了整环共轭空间单位球面的端点，并利用它证明了整环的满射线性等距是由紧Hausdorff空间的同胚诱导的加权复合算子。当基础拓扑空间满足第一可数性公理时，将这一结果推广到二阶导子，进一步研究了整环上的小有界同构，证明了如果存在小有界同构，则基础拓扑空间是同胚的.作为副产品，我们得到了真实的直线的单位区间上连续可微函数空间上的Korovkin型逼近定理.

项目成果

Go Hirasawa: "Korovkin type approximation theorems on the space of continuously differentiable functions"Approximation theory and its applications. in press.

平泽刚：《连续可微函数空间上的科洛夫金型逼近定理》逼近理论及其应用。

Toshiko Matsumoto: "Surjective linear isometries of the domain of a*-derivation equipped with the Cambern norm"Mathematische Zeitshrift. 230. 185-200 (1999)

Toshiko Matsumoto：“配备 Cambern 范数的 a* 导数域的满射线性等距”Mathematische Zeitshrift。

T.Matsumoto: "Suvjective linear isometries of the domain of a *-derivation equipped with the Cambern norm" Mathematische Zeitschrift. 230. 185-200 (1999)

T.Matsumoto：“配备坎本范数的*-导数域的主观线性等距”Mathematicische Zeitschrift。

Toshiko Matsumoto: "Surjective linear isometrics of the domain of a *-derivation equipped with the Cambern norm"Mathematische Zeitshrift. 230. 185-200 (1999)

Toshiko Matsumoto：“配备坎伯范数的*-导数域的满射线性等距”Mathematische Zeitsrift。

Toshiko Matsumoto: "Small bound isomorphisms of the domain of a closed *-derivation"International Journal of Mathematics and Mathematical Sciences. in press.

Toshiko Matsumoto：“封闭*-导数域的小界同构”国际数学与数学科学杂志。