A Perturbation Problem for the Shift Semigroup

An E0-semigroup is a semigroup of unital endomorphisms of B(H), the set of bounded operators on an infinite dimensional separable Hilbert space H , with appropriate continuity. Despite the fact that B(H) is the simplest infinite dimensional factor, the classification of E0-semigroups is far from a satisfactory stage mainly caused by the presence of so called type II and type III examples [11],[12],[17]. Indeed, these classes are related to various areas of analysis, such as probability theory and harmonic analysis, which makes the subject more interesting and worth investigating than one would expect at first sight. The reader is referred to Arveson’s monograph [3] and contributions in [13], in particular those of Arveson, Powers, and Tsirelson among others, for history and basic results for the subject. In this paper, we investigate a purely operator theoretical problem motivated by Tsirelson’s construction of uncountably many mutually non-cocycle conjugate E0semigroups of type III [18]. Namely, let {St}t≥0 be the shift semigroup of L2(0,∞), that is