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


Spectral theorem for unbounded strongly continuous groups on a Hilbert space

Authors: Khristo Boyadzhiev and Ralph deLaubenfels
Journal: Proc. Amer. Math. Soc. 120 (1994), 127-136
MSC: Primary 47D03; Secondary 47A60
MathSciNet review: 1186983
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Abstract: Suppose $ B$ is a closed, densely defined linear operator on a Hilbert space and $ a > 0$. Denote $ \{ z\vert\vert\operatorname{Im} (z)\vert < b\} $ by $ {H_b}$.

We show that $ B$ has an $ {H^\infty }({H_b})$ functional calculus, for all $ b > a$, if and only if $ iB$ generates a strongly continuous group of operators of exponential type $ a$. We obtain specific upper bounds on $ \vert\vert f(B)\vert\vert$, in terms of $ \sup \{ {e^{ - b\vert t\vert}}\vert\vert{e^{itB}}\vert\vert\;\vert t \in {\mathbf{R}}\} $.

Corollaries include the spectral theorem for closed operators on a Hilbert space and a generalization of a result due to McIntosh relating imaginary powers and $ {H^\infty }$ functional calculi.

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PII: S 0002-9939(1994)1186983-0
Article copyright: © Copyright 1994 American Mathematical Society