Averages of operators and their positivity

Nagisa, Masaru; Wada, Shuhei

doi:10.1090/S0002-9939-98-04070-2

Averages of operators and their positivity
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by Masaru Nagisa and Shuhei Wada PDF

Proc. Amer. Math. Soc. 126 (1998), 499-506 Request permission

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 \[ \|I- {\frac {1 }{N+1 }}\sum \limits _{i=k}^{k+N} T^{i} \|, \|I- {\frac {1 }{N+2 }}\sum \limits _{i=k}^{k+N+1} T^{i} \| \le 1\] for any non-negative integer $k$. For several commuting operators, we can extend this result and get the similar statement.

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