Abstract: We discuss representations of product systems (of W-correspondences) over the semigroup of n-tuple of non-negative integers and show that, under certain pureness and Szego positivity conditions, a completely contractive representation can be dilated to an isometric representation. For n = 1, 2 this is known to hold in general (without assuming the conditions) but, for n ≥ 3, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria and Sarkar. Our dilation is explicitly constructed and we present some applications.

Abstract: Given a bounded operator Q on a Hilbert space H, a pair of bounded operators (T, S) on H is said to be Q-commuting if TS = QST or TS = SQT, or TS = STQ. We give an explicit construction of isometric dilations for pairs of Q-commuting contractions for unitary Q, which generalizes the isometric dilation of Ando for pairs of commuting contractions. In particular, for Q = q_H, where q is a complex number of modulus 1, this gives, as a corollary, an explicit construction of isometric dilations for pairs of q-commuting contractions which are well studied. There is an extended notion of q-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an n-tuple of q-commuting contractions, where n  3. Generalizing the class of commuting contractions considered by Brehmer, we construct a class of n-tuples of q-commuting contractions and nd isometric dilations explicitly for the class.

Abstract: In this article, we study commutant lifting theorems, more generally intertwining lifting theorems, for weighted Bergman spaces over the unit ball in the n-dimensional complex plane. In the particular case of the Hardy space over the unit disc and the Drury–Arveson space over the unit ball, our commutant lifting theorem provides an alternative proof of the classical commutant lifting theorems of Sarason and Ball, Trent and Vinnikov.

Abstract: It is well-known that an n-tuple (n ≥ 3) of commuting contractions does not posses an isometric dilation, in general. Considering a class of n-tuple of commuting contractions satisfying certain positivity assumption, we construct their isometric dilations and consequently establish their von Neumann inequality. The positivity assumption is related to Brehmer positivity and motivated by the study of isometric dilations of operator tuples in Barik et al. (Trans Amer Math Soc 372(2):1429–1450, 2019).

Abstract: Motivated by Ball, Li, Timotin and Trent’s Schur-Agler class version of commutant lifting theorem, we introduce a class of n-tuples of commuting contractions on a Hilbert space H. We always assume that n ≥3. The importance of this class of n-tuples stems from the fact that the von Neumann inequality or the existence of isometric dilation does not hold in general for n-tuples, n ≥3, of commuting contractions on Hilbert spaces (even in the level of finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, we prove that tuples in that class always admit explicit isometric dilations and satisfy a refined von Neumann inequality in terms of algebraic varieties in the closure of the unit polydisc in the n-dimensional complex plane.

Abstract: The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in C[z] or C[z_1, z_2], respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for n-tuples, n ≥ 3, of commuting contractions. The goal of this paper is to provide a taste of isometric dilations, von Neumann inequality, and a refined version of von Neumann inequality for a large class of n-tuples, n ≥ 3, of commuting contractions.