In this paper we study the question whether 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 . However, we construct a contraction semigroup with growth bound minus infinity for which 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc75-0-18,
author = {Hans Zwart},
title = {Is $A^{-1}$ an infinitesimal generator?},
journal = {Banach Center Publications},
volume = {75},
year = {2007},
pages = {303-313},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc75-0-18}
}
Hans Zwart. Is $A^{-1}$ an infinitesimal generator?. Banach Center Publications, Tome 75 (2007) pp. 303-313. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc75-0-18/