Cyclically monotone linear operators
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- by Elias S. W. Shiu PDF
- Proc. Amer. Math. Soc. 59 (1976), 127-132 Request permission
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 $|\operatorname {Arg} (Tx,x)| \leqslant \pi /n,\forall x \in \mathcal {H}$. 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 $\{ z \in {\mathbf {C}}:|\operatorname {Arg} \;z| \leqslant \pi /m + \pi /n\}$.References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 59 (1976), 127-132
- MSC: Primary 47A10; Secondary 47B44
- DOI: https://doi.org/10.1090/S0002-9939-1976-0410417-3
- MathSciNet review: 0410417