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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Semigroups of scalar type operators on Banach spaces

Author: Ahméd Ramzy Sourour
Journal: Trans. Amer. Math. Soc. 200 (1974), 207-232
MSC: Primary 47D05
MathSciNet review: 0365228
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Abstract: The main result is that if $ \{ T(t):t \geqslant 0\} $ is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space $ X$ and if the resolutions of the identity for $ T(t)$ are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure $ K( \cdot )$ such that $ T(t) = \smallint \exp (\lambda t)dK(\lambda )$, for $ t \geqslant 0$. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that $ R$ is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.

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Keywords: Semigroup of operators, spectral operator, scalar type operator, infinitesimal generator
Article copyright: © Copyright 1974 American Mathematical Society