Geodesic
Duality in spectral optimization and numerical ranges of a family of self-adjoint operators
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 189-192
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It is shown that the main condition under which the duality relation is valid in the problems of spectral optimization is related to the numerical ranges of a family of self-adjoint operators. 
@article{ZNSL_1982_107_a12,
     author = {Yu. Sh. Abramov},
     title = {Duality in spectral optimization and numerical ranges of a~family of self-adjoint operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {189--192},
     year = {1982},
     volume = {107},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_107_a12/}
}
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Yu. Sh. Abramov. Duality in spectral optimization and numerical ranges of a family of self-adjoint operators. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 189-192. http://geodesic.mathdoc.fr/item/ZNSL_1982_107_a12/