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Journal of Operator Theory

Volume 46, Issue 1, Summer 2001  pp. 139-157.

Extensions of semigroups of operators

Authors Charles J.K. Batty (1), and Stephen B. Yeates (2)
Author institution: (1) St. John's College, Oxford OX1 3JP, England
(2) St. John's College, Oxford OX1 3JP, England

Summary:  Let $T$ be a representation of an abelian semigroup $S$ on a Banach space $X$. We identify a necessary and sufficient condition, which we name superexpansiveness, for $T$ to have an extension to a representation $U$ on a Banach space $Y$ containing $X$ such that each $U(t)$ $(t \in S)$ has a contractive inverse. Although there are many such extensions $(Y,U)$ in general, there is a unique one which has a certain universal property. The spectrum of this extension coincides with the unitary part of the spectrum of $T$, so various results in spectral theory of group representations can be extended to superexpansive representations.

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