Geodesic
On the Inverse of the Generator of a Bounded $C_0$-Semigroup
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 6-12

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Let $A$ be the generator of a uniformly bounded $C_0$-semigroup in a Banach space $B$, and let $A$ have a densely defined inverse $A^{-1}$. We present sufficient conditions on the resolvent $(A-\lambda I)^{-1}$, $\operatorname{Re}\lambda>0$, under which $A^{-1}$ is also the generator of a uniformly bounded $C_0$-semigroup.
Keywords: uniformly bounded $C_0$-semigroup, inverse of the generator, Banach space, Carleson embedding theorem. 
@article{FAA_2004_38_4_a1,
     author = {A. M. Gomilko},
     title = {On the {Inverse} of the {Generator} of a {Bounded} $C_0${-Semigroup}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {6--12},
     publisher = {mathdoc},
     volume = {38},
     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_4_a1/}
}
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A. M. Gomilko. On the Inverse of the Generator of a Bounded $C_0$-Semigroup. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 6-12. http://geodesic.mathdoc.fr/item/FAA_2004_38_4_a1/