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

Cyclically monotone linear operators


Author: Elias S. W. Shiu
Journal: Proc. Amer. Math. Soc. 59 (1976), 127-132
MSC: Primary 47A10; Secondary 47B44
MathSciNet review: 0410417
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Abstract: A linear operator on a complex Hilbert space $ \mathcal{H}$ is called $ n$-cyclically monotone if for each sequence $ {x_0},{x_1}, \ldots ,{x_{n - 1}},{x_n} = {x_0}$ of $ n$ elements in $ \mathcal{H},\Sigma _{j = 0}^{n - 1}\operatorname{Re} (T{x_j} - {x_{j + 1}}) \geqslant 0$. We show that $ T$ is $ n$-cyclically monotone if and only if $ \vert\operatorname{Arg} (Tx,x)\vert \leqslant \pi /n,\forall x \in \mathcal{H}$. If $ {T_m}$ and $ {T_n}$ are $ m$- and $ n$-cyclically monotone operators, then the spectrum of the product $ {T_m}{T_n}$ lies in the sector $ \{ z \in {\mathbf{C}}:\vert\operatorname{Arg} \;z\vert \leqslant \pi /m + \pi /n\} $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0410417-3
PII: S 0002-9939(1976)0410417-3
Keywords: Cyclically monotone operators, numerical ranges, tensor products, spectra of products
Article copyright: © Copyright 1976 American Mathematical Society