Geodesic
Some analogues of von-Heumann's inequality for $J$-contractions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 165-172 
M. M. Malamud. Some analogues of von-Heumann's inequality for $J$-contractions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 165-172. http://geodesic.mathdoc.fr/item/ZNSL_1987_157_a16/
@article{ZNSL_1987_157_a16,
     author = {M. M. Malamud},
     title = {Some analogues of {von-Heumann's} inequality for $J$-contractions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {165--172},
     year = {1987},
     volume = {157},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_157_a16/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $J$ be a self-adjoint operator satisfying $J^2=I$. We prove that for any $J$-contraction $T$ (i. e. $T^*JT-J\leqslant0$) and any inner function $f$ holomorphic on the spectrum of $T$ the function $f(T)$ is a $J$-contraction too. It is also proved that for $J\ne\pm I$ only inner functions $f$ satisfy this property. We consider other analogues of von-Neumann's inequality.