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acting on a Hilbert space H with T = ∏ n i=1 T i , we find a necessary and sufficient condition under which (T 1 , . . . , T n ) dilates to commuting isometries (V 1 , . . . ,V n ) on the minimal isometric dilation space T , where V = ∏ n i=1 V i is the minimal isometric dilation of T . We construct both Schaffer and Sz. Nagy-Foias type isometric and unitary dilations for (T 1 , . . . , T n ) on the minimal dilation spaces of T . Also, a different dilation is constructed when the product T is a C. 0 contraction, that is T * n → 0 as n → ∞. As a consequence of these dilation theorems we obtain different functional models for (T 1 , . . . , T n ) in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are analytic functions in one variable. The dilation when T is C. 0 leads to a conditional factorization of a T . Several examples have been constructed. CONTENTSWe show an explicit Sch äffer type construction of such an isometric dilation on K = H ⊕ l 2 (D T ), where D T = Ran(I − T * T ) 1 2 (see Theorem 3.4). We also show in Theorem 3.7 that such a dilation can be constructed with the conditions (1) − (4) only though we do not have an exact converse part then. We place a special emphasis on the case when the product T is a C. 0 contraction, i.e. T * n → 0 strongly as n → ∞. We show Theorem 4.1 that an analogue of Theorem 1.1 can be achieved for the C. 0 case with a weaker hypothesis. We explicitly construct isometric dilation in

Consider a nonzero contraction T and a bounded operator X satisfying T X = qXT for a complex number q. There are some interesting results in the literature on q-commuting dilation and q-commutant lifting of such pair (T, X) when |q| = 1. Here we improve a few of them to the class of scalars q satisfying |q| ≤ 1 T .

acting on a Hilbert space H with T = ∏ n i=1 T i , we show that (T 1 , . . . , T n ) dilates to commuting isometries (V 1 , . . . ,V n ) on the minimal isometric dilation space of T with V = ∏ n i=1 V i being the minimal isometric dilation of T if and only ifi=1 Y i being the minimal isometric dilation of T * . Then, we prove an analogue of this result for unitary dilations of (T 1 , . . . , T n ) and its adjoint. We find a necessary and sufficient condition such that (T 1 , . . . , T n ) possesses a unitary dilation (W 1 , . . . ,W n ) on the minimal unitary dilation space of T with W = ∏ n i=1 W i being the minimal unitary dilation of T . We show an explicit construction of such a unitary dilation on both Sch äffer and Sz. Nagy-Foias minimal unitary dilation spaces of T . Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when T is a C. 0 contraction, i.e. when T * n → 0 strongly as n → ∞. We construct a different unitary dilation for (T 1 , . . . , T n ) when T is a C. 0 contraction.