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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Inverses of generators


Author: Ralph deLaubenfels
Journal: Proc. Amer. Math. Soc. 104 (1988), 443-448
MSC: Primary 47D05; Secondary 47A60
MathSciNet review: 962810
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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

$\displaystyle {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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0962810-6
PII: S 0002-9939(1988)0962810-6
Article copyright: © Copyright 1988 American Mathematical Society