Lower bounds for unbounded operators and semigroups

Let $A$ be an unbounded operator on a Banach space $X$. It is sometimes useful to improve the operator $A$ by extending it to an operator $B$ on a larger Banach space $Y$ with smaller spectrum. It would be preferable to do this with some estimates for the resolvent of $B$, and also to extend bounded operators related to $A$, for example a semigroup generated by $A$. When $X$ is a Hilbert space, one may also want $Y$ to be Hilbert space. Results of this type for bounded operators have been given by Arens, Read, M\"uller and Badea, and we give some extensions of their results to unbounded operators and we raise some open questions. A related problem is to improve properties of a $C_0$-semigroup satisfying lower bounds by extending it to a $C_0$-group on a larger space or by finding left-inverses. Results of this type for Hilbert spaces have been obtained by Louis and Wexler, and by Zwart, and we give some additional results.