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


Zwart, H.J. (2007) Is $A^{-1}$ an infinitesimal generator? Banach center publications, 75 . pp. 303-313. ISSN 0137-6934

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Abstract:In this paper we study the question whether $A^{-1}$ is the infinitesimal generator of a bounded $C_0$-semigroup if $A$ generates a bounded $C_0$-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 1988 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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Electrical Engineering, Mathematics and Computer Science (EEMCS)
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