Decompositions of Beurling Type for E0-semigroups

We define tensor product decompositions of E0-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent. Introduction. A well-known theorem of Beurling characterizes invariant subspaces of the right shift on l(N) by inner functions in the unit disc. In this case the restriction of the right shift to a nontrivial invariant subspace is automatically conjugate (unitarily equivalent) to the original shift. This interesting self-similar structure is fundamental in many respects. It is the prototype of a very fruitful interaction between operator theory and function theory, see for example [Ni, FF]. In this paper we want to study a somewhat analogous self-similar structure for operators on a different level. While the original setting concerns isometries and decompositions of the Hilbert space into direct sums, we want to study E0-semigroups, i.e., pointwise weak -continuous semigroups of unital ∗-endomorphisms of B(H) for some complex separable Hilbert space H (cf. [Ar]), and decompositions of the Hilbert space into tensor products. To make the analogy visible, we present in Section 1 the Nagy-Foiaş functional model for ∗-stable contractions and their characteristic functions in a suitable way and in particular we emphasize a limit formula for the characteristic function which is not made explicit in the standard presentations. This analogy motivates the definition of decompositions of Beurling type for E0-semigroups in Section 2. It is then shown that there is a reformulation in terms of unitary cocycles for amplifications of the E0-semigroup. Relevant properties of the cocycles are adaptedness and exactness. In fact, if the E0-semigroup is a CCR-flow (cf. [Ar]), then this leads to a setting which has been extensively studied by quantum probabilists (cf. [Pa]). We have a tensor product of an initial Hilbert space with a symmetric Fock space over L[0,∞), maybe with a multiplicity space 2000 Mathematics Subject Classification: Primary 46L55, 47D03; Secondary 81S25.