Spectral theorem for unbounded strongly continuous groups on a Hilbert space

Abstract

Suppose B B is a closed, densely defined linear operator on a Hilbert space and a > 0 a > 0 . Denote { z | | Im ⁡ ( z ) | > b } \{ z||\operatorname {Im} (z)| > b\} by H b {H_b} . We show that B B has an H ∞ ( H b ) {H^\infty }({H_b}) functional calculus, for all b > a b > a , if and only if i B iB generates a strongly continuous group of operators of exponential type a a . We obtain specific upper bounds on | | f ( B ) | | ||f(B)|| , in terms of sup { e − b | t | | | e i t B | | | t ∈ R } \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 ∞ {H^\infty } functional calculi.