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## Banach Center Publications

2007 | 75 | 1 | 303-313
Tytuł artykułu

### Is $A^{-1}$ an infinitesimal generator?

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In this paper we study the question whether $A^{-1}$ is the infinitesimal generator of a bounded C₀-semigroup if A generates a bounded C₀-semigroup. If the semigroup generated by A is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^{-1}$. However, we construct a contraction semigroup with growth bound minus infinity for which $A^{-1}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its inverse does not generate a semigroup. Hence we show that the question posed by deLaubenfels in [13] must be answered negatively. All these examples are on Banach spaces. On a Hilbert space the question whether the inverse of a generator of a bounded semigroup also generates a bounded semigroup still remains open.
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Tom
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303-313
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2007
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• Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
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