Abstract
The paper contains a complete exposition and proofs of our results on the theory of sufficient collections of norms. Consider a uniformly bounded irreducible semigroup G of operators in a finite dimensional real linear space V. We consider G-contractive norms on V, i.e., such norms that every operator from G is contractive with respect to each of them. A collection of G-contractive norms { ∥ · ∥α }α∊A is called sufficient if for any linear operator T : V → V the following implication holds: if T is contractive in every norm ‖ · ‖α (α ∊ A) then T is contractive in every G-contractive norm. We describe all sufficient collections, construct two canonical sufficient collections and study their extremal properties.
The research was supported in part by a grant from the Ministry of Absorption and the Rashi Foundation.
The research was supported in part by a grant from the Ministry of Science and the «Maagara» — a special project for absorption of new immigrants, at the Department of Mathematics, Technion.
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References
Bourbaki, N.: Espaces Vectoriels Topologiques, Herman, Paris, 1968.
Calderon, A.P.: Spaces between L 1 and L ∞ and the theorem of Marcinkiewicz, Studia Math., 26(1966), 229–273.
Leichtweiss, K.: Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin, (1980).
Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and its Applications, Mathematics in Science and Engineering, 143(1979).
Mityagin, B.S.: An interpolation theorem for modular spaces, Matem. Sbornik, v. 66, 4(1965), 473–482 (Russian).
Veselova, L.V., Zobin, N.M.: A general theory of interpolation and duality, in «Constructive theory of functions and funct. analysis» (edited by A.N. Sherstnev), Kazan University Press, (1990), 14–29 (Russian).
Zobin, N.M.: On extreme mixed norms, Uspekhi Mat. Nauk, 39(1984), 157–158 (Russian).
Zobin, N.M., Zobina, V.G.: Interpolation in spaces possessing the prescribed symmetries, Funct. Anal. Pril., v. 12, 4(1978), 85–86 (Russian).
Zobin, N.M., Zobina, V.G.: Interpolation in the spaces with prescribed symmetries. Finite dimension, J. Izvestiya vusov, Matematika, 4(1981), 20–28 (Russian).
Zobin, N.M., Zobina, V.G.: Minimal sufficient collections for semigroups of operators, J. Izvestiya vusov, Matematika, 11(1989), 31–35 (Russian).
Zobin, N.M., Zobina, V.G.: Duality in operator spaces and problems of interpolation of operators, Pitman Research Notes in Math., Longman Sci. & Tech., London, v. 257 (1992), 123–144.
Zobin, N., Zobina, V.: Coxeter groups and interpolation of operators, Preprint Max-Planck-Inst. für Math. — Bonn, MPI/93–39 (1993), 1–44.
Zobina, V.G.: Interpolation in spaces with prescribed symmetries and the uniqueness of sufficient collections, Soobshenya AN Gruzin. SSR, v. 93, 2(1979), 301–303 (Russian).
Zobina, V.G.: Duality in interpolation of operators, Soobshenya AN Gruzin. SSR, V. 95, 1(1979), 45–48 (Russian).
Zobina, V.G.: Interpolation in spaces with prescribed symmetries. Group B2 ⊗ B2, preprint VINITI, 2.08.78, 2828–78(1978), (Russian).
Zobina, V.G.: Interpolation of operators in spaces with a Coxeter group as the group of symmetries, preprint VINITI 16.02.82, 3318–82(1982), (Russian).
Zobina, V.G.: Sufficient collections for semigroups of operators, J. Izvestiya vusov, Matematika, 1(1989), 33–35 (Russian).
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Zobin, N., Zobina, V. (1994). A General Theory of Sufficient Collections of Norms with a Prescribed Semigroup of Contractions. In: Feintuch, A., Gohberg, I. (eds) Nonselfadjoint Operators and Related Topics. Operator Theory: Advances and Applications, vol 73. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8522-5_16
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DOI: https://doi.org/10.1007/978-3-0348-8522-5_16
Publisher Name: Birkhäuser, Basel
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