Cyclically monotone linear operators

Shiu, Elias S. W.

doi:10.1090/S0002-9939-1976-0410417-3

Cyclically monotone linear operators
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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

Haim Brezis and Felix E. Browder, Maximal monotone operators in nonreflexive Banach spaces and nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 81 (1975), 82–88. MR 407670, DOI 10.1090/S0002-9904-1975-13645-7
Arlen Brown and Carl Pearcy, Spectra of tensor products of operators, Proc. Amer. Math. Soc. 17 (1966), 162–166. MR 188786, DOI 10.1090/S0002-9939-1966-0188786-5
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Peter Hess, On nonlinear equations of Hammerstein type in Banach spaces, Proc. Amer. Math. Soc. 30 (1971), 308–312. MR 282268, DOI 10.1090/S0002-9939-1971-0282268-X
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497–510. MR 193549

Convex analysis

43

Teishirô Saitô, Numerical ranges of tensor products of operators, Tohoku Math. J. (2) 19 (1967), 98–100. MR 229076, DOI 10.2748/tmj/1178243352
Survey of numerical analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR 0135221
Helmut Wielandt, On the eigenvalues of $A+B$ and $AB$, J. Res. Nat. Bur. Standards Sect. B 77B (1973), 61–63. MR 335537
James P. Williams, Spectra of products and numerical ranges, J. Math. Anal. Appl. 17 (1967), 214–220. MR 203491, DOI 10.1016/0022-247X(67)90146-1