Spectral theorem for unbounded strongly continuous groups on a Hilbert space

Boyadzhiev, Khristo; deLaubenfels, Ralph

doi:10.1090/S0002-9939-1994-1186983-0

Spectral theorem for unbounded strongly continuous groups on a Hilbert space
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Proc. Amer. Math. Soc. 120 (1994), 127-136 Request permission

Abstract:

Suppose $B$ is a closed, densely defined linear operator on a Hilbert space and $a > 0$. Denote $\{ z||\operatorname {Im} (z)| < b\}$ 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 $||f(B)||$, in terms of $\sup \{ {e^{ - b|t|}}||{e^{itB}}||\;|t \in {\mathbf {R}}\}$. 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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