1999 Fiscal Year Final Research Report Summary
Synthetic Study for Real Analysis and its Applications
| Project/Area Number |
10440048
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | OKAYAMA UNIVERSITY |
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Principal Investigator |
SATO Ryotaro Faculty of Science, Okayama University, Professor, 理学部, 教授 (50077913)
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| Co-Investigator(Kenkyū-buntansha) |
KITA Hiro-o Oita University, Faculty of Education, Professor, 教育福祉科学部, 教授 (20224941)
TANAKA Naoki Faculty of Science, Okayama University, Associate Professor, 理学部, 助教授 (00207119)
HASEGAWA Shigeru Shibaura Institute of Technology, Faculty of Technology, Professor, 工学部, 教授 (50052832)
TAKAHASHI Yasuji Okayama Prefectural University, Faculty of Computer Science and System Engineering, Professor, 情報工学部, 教授 (30001853)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Ergodic theorems / positive operators / maximal ergodic functions / measure preserving transformations / semigroups of operators / almost everywhere convergence |
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| Research Abstract |
We considered pointwise ergodic thorems for n-dimensional semigroups of operators on vector-valued functions spaces. It is difficult to prove pointwise ergodic theorems for these semigroups. The main reason is that the estimate of the maximal ergodic functions is not obtained (except for the one-dimensional semigroup case). This remains still an open problem. But, in case the semigroup consists of positive operators, the estimate can be obtained by using the reduction method of Dunford and Schwartz. This method is applied to complex-valued function spaces, and Emilion has succeeded to prove a general local ergodic theorem by this method. Unfortunately, this is not applied to vector-valued function spaces, because the linear modulus of an operator does not exists in general for an operator on vector-valued function spaces. About this problem, we extended the notion of linear modulus of an operator, and then proved, under some suitable norm conditions on operator, that the pointwise ergodic theorem holds for n-dimensional semigroups of operators. Also we proved that this method remains valid to prove the local ergodic theorem for n-dimensional semigroups of operators on vector-valued functions spaces.
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