Studies of the Structure of Operators on Function Spaces
Project/Area Number  10640189 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  Niigata Institute of Technology 
Principal Investigator 
WATANABE Seiji Faculty of Engineering,Kanazawa Institute of Tecnology,Professor, 工学部, 教授 (40018271)

CoInvestigator(Kenkyūbuntansha) 
TAKENO Sigeharu 新潟工科大学, 工学部, 助教授 (30251789)
TANAKA Kensuke 新潟工科大学, 工学部, 教授 (70018258)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥600,000 (Direct Cost : ¥600,000)

Keywords  Function Spaces / Linear Operators / Unbounded Derivations / Weighted Compositi Operators / 関数空間 / 線形作用素 / 非有界微分 / 荷重合成作用素 
Research Abstract 
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 smallbound 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 smallbound isomorphisms on the domain and showed that if there is a small boundisomorphism, then the underlying topological spaces are homeomorphic. As byproduct, we obtained Korovkin type approximation theorems on the space of continuously differentiable functions on the unit interval of the real line.

Report
(4results)
Research Output
(11results)