Spectral theory for one-parameter groups of isometries

Abstract

If { U( t)} t∈ R is a given unitary one-parameter group of operators in Hilbert space, there are well-known L p -integrability conditions ( P ⩾ 1) on the imaginary part of the resolvent operators which imply absolute continuity of the spectral resolution for { U( t)}. Although one-parameter groups of isometrics in Banach space do not in general have spectral resolutions, they still have an attractive spectral theory. We shall display resolvent type conditions which imply spectral continuity for such one-parameter groups, thus generalizing the Hilbert space results. The results are applied to the case of one-parameter groups of automorphisms of C ∗-algebras in the last section.