The central intertwining lifting and strict contractions

Abstract

Abstract. In this paper we give a necessary and sufficient condition for the central intertwining lifting of a strict contraction to be strictly contractive. As an application, we obtain a factorization of D −2Ac when the central intertwining lifting Ac of A is a strict contraction.Let H, H be (complex, separable) Hilbert spaces, T and T contractions in L(H) and L(H ), respectively, having minimal isometric dilations U and U in L(K) and L(K ). If A is a contraction operator in L(H, H ) such that AT = T A, the celebrated Sz.-Nagy-Foias commutant lifting theorem ([3, 7, 8] | kerT * is invertible (see [5]). For the convenience of the reader we begin by reviewing some notation and terminology (see [3,4,5,6,8] . Since the minimal isometric dilation of T is unique (up to an isomorphism (see [3,8])), we may and do assume that U ∈ L(K ), the minimal isometric dilation of T is the Sz.-Nagy-Schäffer minimal isometric dilation of T , i.e. K = H ⊕H 2 (D T ) and U