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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Averages of operators and their positivity


Authors: Masaru Nagisa and Shuhei Wada
Journal: Proc. Amer. Math. Soc. 126 (1998), 499-506
MSC (1991): Primary 47B65; Secondary 47B44
MathSciNet review: 1415335
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Abstract: Let $T$ be a bounded linear operator on a Hilbert space. We prove that $T$ is positive, if there exists a positive integer $N$ such that

\begin{displaymath}\|I- {\frac{1 }{N+1 }}\sum \limits _{i=k}^{k+N} T^{i} \|, \|I- {\frac{1 }{N+2 }}\sum \limits _{i=k}^{k+N+1} T^{i} \| \le 1\end{displaymath}

for any non-negative integer $k$. For several commuting operators, we can extend this result and get the similar statement.


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Additional Information

Masaru Nagisa
Affiliation: Department of Mathematics and Informatics, Faculty of Science, Chiba University 1-33 Yayoi-cho, Inage-ku Chiba, 263, Japan
Email: nagisa@math.s.chiba-u.ac.jp

Shuhei Wada
Affiliation: Department of Information and Computer Engineering, Kisarazu National College of Technology 2-11-1 Kiyomidai-Higashi, Kisarazu, Chiba, 292, Japan
Email: wada@gokumi.j.kisarazu.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04070-2
PII: S 0002-9939(98)04070-2
Received by editor(s): April 30, 1996
Received by editor(s) in revised form: August 12, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society