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Extremal properties of contraction semigroups onc 0

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Abstract

For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C 0) contraction semigroup (T t ) t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx *∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T t ) t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc 0. In this article, we show the answer is negative

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Authors and Affiliations

  1. Department of Mathematics, Memphis State University, 38152, Memphis, TN

    Pei-Kee Lin

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  1. Pei-Kee Lin
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Communicated by J. A. Goldstein

Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to thank Profs. Goldstein and Jamison for their valuable suggestions.

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Lin, PK. Extremal properties of contraction semigroups onc 0 . Semigroup Forum 53, 208–211 (1996). https://doi.org/10.1007/BF02574135

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  • DOI: https://doi.org/10.1007/BF02574135

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