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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