Inverses of generators

deLaubenfels, Ralph

doi:10.1090/S0002-9939-1988-0962810-6

Inverses of generators
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by Ralph deLaubenfels PDF

Proc. Amer. Math. Soc. 104 (1988), 443-448 Request permission

Abstract:

Let $A$ be a (possibly unbounded) linear operator on a Banach space $X$ that generates a bounded holomorphic semigroup of angle $\theta (0 < \theta \leq \pi /2)$. We show that, if the range of $A$ is dense, then $A$ is one-to-one, and ${A^{ - 1}}$ (defined on the range of $A$) generates a bounded holomorphic semigroup of angle $\theta$, given by \[ {e^{z{A^{ - 1}}}} = \int {{e^{ - w}}{{(wA + z)}^{ - 1}}\frac {{dw}}{{2\pi i}},} \] over an appropriate curve. When $X$ is reflexive, it is sufficient that $A$ be one-to-one.

References

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Generators of strongly continuous semigroups