Adjoints of semigroups acting on vector-valued function spaces |
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Authors: | G. Greiner J. M. A. M. van Neerven |
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Affiliation: | 1. Universit?t Tübingen, Auf der Morgenstelle 10, D-7400, Tübingen, Germany 2. Centre for Mathematics and Computer Science, P.O. Box 4079, 1009, AB Amsterdam, The Netherlands
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Abstract: | LetT(t) be the translation group onY=C 0(ℝ×K)=C 0(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ⊙=L 1(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT 0(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT 0(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C 0(Ω;X) andL p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support. |
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