Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. Cox, R. H., Matrices all whose powers lie close to the identity, Amer. Math. Monthly 73 (1966), 813.
  • 2. DePrima, C. R., Richard, B. K., A Characterrization of the Positive Cone of B(${\mathcal{H}}$), Indiana Univ. Math. Jour. 23 (1973), 1085-1087. MR 47:4052
  • 3. Halmos, P. R., A Hilbert Space Problem Book Second Ed., Springer-Verlag, New York, 1982. MR 36:5743
  • 4. Kato T., Some mapping theorems for the numerical range, Proc. Japan Acad. 41 (1965), 652-655.
  • 5. Nagisa M., Wada S., An extension of the mean ergodic theorem, to appear in Math. Japon. (Vol. 48, No. 1).
  • 6. Nakamura M., Yoshida M., On a generalization of a theorem of Cox, Proc. Japan Acad. 43 (1967), 108-110. MR 36:708
  • 7. Takesaki, M., Theory of Operator Algebras I, Springer-Verlag, 1979. MR 81e:46038

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47B65, 47B44

Retrieve articles in all journals with MSC (1991): 47B65, 47B44

Additional Information

Masaru Nagisa
Affiliation: Department of Mathematics and Informatics, Faculty of Science, Chiba University 1-33 Yayoi-cho, Inage-ku Chiba, 263, Japan

Shuhei Wada
Affiliation: Department of Information and Computer Engineering, Kisarazu National College of Technology 2-11-1 Kiyomidai-Higashi, Kisarazu, Chiba, 292, Japan

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

American Mathematical Society