Commutant lifting theorem for n-tuples of contractions

We show that the commutant lifting theorem for n-tuples of commuting contractions with regular dilations fails to be true. A positive answer is given for operators which ”double intertwine” given n-tuples of contractions. The commutant lifting theorem is one of the most important results of the Sz. Nagy—Foias dilation theory. It is usually stated in the following way: Theorem. Let T and T ′ be contractions in Hilbert spaces H and H ′. Let A : H → H ′ be a contraction which intertwines T and T ′, i.e. AT = T ′A. Let V ∈ B(K+) and V ′ ∈ B(K ′ +) be the minimal isometric dilations of T and T ′. Then there exists a contraction B : K+ → K ′ + such that BV = V ′B and APH = PH′B. (We denote by PM the orthogonal projection onto a closed subspace M). The commutant lifting theorem has been studied intensely (see [3]) because, apart from its interesting operator-theoretic consequences, it has a number of applications concerning interpolation problems, in the control theory and even in some pure technical problems. The aim of this paper is to study the commutant lifting theorem for n-tuples of commuting contractions. It is well-known that in general (for n ≥ 3) n-tuples of commuting contractions have no dilations. A pair of commuting contractions has a (Ando) dilation, but this dilation is not unique and the structure of the corresponding space is rather complicated. By example VII.6.3 of [3] the commutant lifting theorem fails to be true for the Ando dilations. Therefore we restrict ourselves to the case of commuting contractions having a regular dilation, which exhibits many properties similar to the case of a single contraction. We use the method of Timotin [7] which relates the commutant lifting theorem with a problem of finding a positive semidefinite extension of some partial operator-valued matrix. Let I be an index set and let, for each α ∈ I, a Hilbert space Hα be given. Let operators Tα,β : Hβ → Hα be given for all α, β ∈ I. We say that the matrix (Tα,β)α,β∈I is positive semidefinite if ∑ α,β∈I < Tα,βhβ , hα >≥ 0 for every function h : α 7→ hα ∈ Hα with a finite support. The research was supported by the grant No. 119106 of the Czech Academy of Sciences. 1991 Mathematics Subject Classification 47A20