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Approximations of the identity operator by semigroups of linear operators


Author: A. Pazy
Journal: Proc. Amer. Math. Soc. 30 (1971), 147-150
MSC: Primary 47.50
DOI: https://doi.org/10.1090/S0002-9939-1971-0287362-5
MathSciNet review: 0287362
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Abstract: Let $ T(t),t \geqq 0$, be a strongly continuous semigroup of linear operators on a Banach space X. It is proved that if for every $ C > 0$ there exists a $ {\delta _c} > 0$ such that $ \left\Vert {I - T(t)} \right\Vert \leqq 2 - Ct\log (1/t)$ for $ 0 < t < {\delta _c}$ then $ AT(t)$ is bounded for every $ t > 0$. It is shown by means of an example that $ \left\Vert {I - T(t)} \right\Vert \leqq 2 - Ct$ for a fixed C and all $ 0 < t < \delta $ is not sufficient to assure the boundedness of $ AT(t)$ for any $ t \geqq 0$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0287362-5
Keywords: Holomorphic, strongly continuous semigroups of bounded linear transformations
Article copyright: © Copyright 1971 American Mathematical Society

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