On perturbations of the isometric semigroup of shifts on the semiaxis

Abstract

Abstract. Perturbations (r τ t ) t≥0 of the semigroup of shifts (τ t ) t≥0 on L 2 (R + ) are studied under the assumption that r τ t −τ t belongs to a certain Schatten-von Neumann class S p with p ≥ 1. It is shown that, for the unitary component in the WoldKolmogorov decomposition of the cogenerator of the semigroup (r τ t ) t≥0 , any singular spectral type may be achieved by S 1 -perturbations. An explicit construction is provided for a perturbation with a given spectral type, based on the theory of model spaces of the Hardy space H 2 . Also, it is shown that an arbitrary prescribed spectral type may be obtained for the unitary component of the perturbed semigroup by a perturbation of class S p with p > 1. §1. IntroductionConsider the isometric semigroup (τ t ) t≥0 of shifts on the space L 2 (R + ),In this paper we are concerned with perturbations (r τ t ) of the semigroup (τ t ) satisfying the following properties: (r τ t ) t≥0 is a strongly continuous semigroup of isometric operators on L 2 (R + ); the difference r τ t − τ t belongs to a certain Schatten-von Neumann ideal S p for every t > 0.The central problem considered in this paper is to describe all possible spectral types of perturbed isometric semigroups. The spectral type of a semigroup determines the semigroup uniquely up to unitary equivalence; it is defined by the spectral type of the cogenerator of the group (the definition can be found in §2). For p = 1, the stability of the absolutely continuous spectrum of the unitary dilation implies that the absolutely continuous parts of the cogenerators for the unitary dilations of the semigroups (τ t ) and (r τ t ) are unitarily equivalent (see §2 for the details). The cogenerator of the semigroup (τ t ) is unitarily equivalent to the unilateral shift operator on the Hardy space H 2 . Thus, for p = 1, our problem reduces to the description of all possible singular parts, and we show that any singular type may be realized by some semigroup (r τ t ) t≥0 . For p = 2, in [1,2] it was shown that any spectral type of the unitary component may occur; for the case of a singular spectral measure, a model of such perturbations was constructed. Here we show that similar results are valid for all p > 1.

On applications of the model spaces to the construction of cocyclic perturbations of the semigroup of shifts on the semiaxis

“…A detailed proof of the last statement is given in [5] (proof of Theorem 1.3). To complete the proof it is sufficient to apply Theorem 1.…”

“…Remark. This statement holds true for an arbitrary semigroup (not necessarily being a cocyclic perturbation) of isometric operators (V t , t ≥ 0), if we require that V t − S t ∈ S p , p ≥ 1 (see [5]).…”

“…We should note that the problem of the Markov cocyclic perturbations of the group of unitary operators connected with the questions under consideration is posed in [2], and the Markov cocycles possessing the property W t − I ∈ S 2 , t ≥ 0 are considered in [3,4]. The property (3) was considered in the article [5], where perturbations (V t , t ≥ 0) of semigroup shifts (S t , t ≥ 0) such that V t − S t ∈ S p , p ≥ 1, were investigated. A distinguishing feature of the present paper is that the considered perturbations possess additional cocyclic properties demanding consideration of unitary dilations of semigroups.…”

“…A distinguishing feature of the present paper is that the considered perturbations possess additional cocyclic properties demanding consideration of unitary dilations of semigroups. The technique applied here is analogous to the one in paper [5].…”