On the commutant of asymptotically non-vanishing contractions

One of the main methods of examining non-normal operators, acting on Hilbert spaces, is the theory of contractions. This area of operator theory was developed by Béla Sz.-Nagy and Ciprian Foias from the dilation theorem of Sz.-Nagy. Sz.-Nagy and Foias classi ed the contractions according to their asymptotic behaviour. They got strong structural results in the case when the contraction and its adjoint are simultaneously asymptotically non-vanishing. However, basic questions are still open (e.g. the hyperinvariant and the invariant subspace problems), when only the contraction is asymptotically non-vanishing. In this case one can associate a unitary asymptote to the contraction on a canonical way. The connection with this unitary operator is manifested in an algebra-homomorphism between the commutants of these operators. This commutant mapping is among the few links which relate the contraction to a well-understood operator. It can be exploited to get structure theorems or stability results; see e.g. [Ba], [K3], [K5], [K6], [KL] and [KV]. Hence it is of interest to study its properties. Our purpose was to examine the injectivity of the commutant mapping. One of the results states that this mapping can be injective even in the case when the contraction has a non-trivial stable subspace. Various characterizations of injectivity are provided. Now we give the basic de nitions and x the notation. Let H and K be non-zero, complex, separable Hilbert spaces, and let L(H,K) stand for the system of bounded, linear transformations from H to K. Then L(H) := L(H,H) is the set of operators acting on H. The commutant {T}′ of T ∈ L(H) consists of those operators C ∈ L(H), which commute with T : TC = CT . For any operators A ∈ L(H) and B ∈ L(K), the intertwining set I(A,B) consists of those transformations Y ∈ L(H,K), which satisfy the condition Y A = BY . Any invertible transformation Y ∈ L(H,K) is said to be an a nity. If Y is injective (i.e. kerY = {0}) and has dense range (i.e. (ranY )− = K), then it is called a quasia nity. Let T ∈ L(H) be a contraction: ‖T‖ ≤ 1. The vector h ∈ H is asymptotically vanishing or stable for T if limn→∞ ‖Tnh‖ = 0. The subspace H0 = H0(T ) of all stable vectors is called the stable subspace of T . It is clearly hyperinvariant for T , that is invariant for every operator which commutes with T . We recall that T is of class C0· if T is stable, that is H0 = H. If H0 = {0}, then T is of class C1· and is called asymptotically strongly non-vanishing. T is asymptotically non-vanishing or of class C∗· if H0 6= H. We say that T is of class C·j (j ∈ {0, ∗, 1}) if its adjoint is of class Cj·. Finally the class Cij consists of those operators, which are both in Ci· and C·j (i, j ∈ {0, ∗, 1}). We refer to [SzNF] in connection with the theory of contractions. Unitary asymptotes of operators were studied in several papers, see e.g. [Bea, Chapter XII], [K1], [K2] and [K4]. Here we recall the de nition given in [BK]. The pair (X,W ) is a contractive unitary intertwining pair of the contraction T ∈ L(H) if W ∈ L(K) is unitary and X ∈ I(T,W ) is contractive. We say that (X,W ) is a unitary asymptote of T , if for any other contractive unitary intertwining pair (Y,U) there exists a unique Z ∈ I(W,U) such that Y = ZX and ‖Z‖ ≤ 1. Such a unitary asymptote (X,W ) always exists and is unique up to isomorphism. Given any C ∈ {T}′, the pair (XC,W ) is a unitary intertwining pair for T . By the universality of (X,W ), there exists exactly one D ∈ {W}′ such that XC = DX. Thus we obtain a mapping γ = γT : {T}′ → {W}′, C 7→ D,