Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

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.Statement 1 (On isometric dilations): Let T ∈ T n (H). Then there exist a Hilbert space K(⊇ H) and an n-tuple of commuting isometries V ∈ T n (K) such that T dilates to V . Now any n-tuple of commuting isometries V ∈ T n (K) can be extended to an n-tuple of commuting unitaries, that is, there exist a Hilbert space L containing K and an n-tuple of commuting unitary operators U ∈ T n (L) which extends V [24]. Therefore, the celebrated von Neumann inequality is an immediate consequence of Statement 1 (cf. [24]):

Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications

“…We conclude the latter section by applying our results to obtain a characterization for pure contractions belonging to the commutant of certain tuples of commuting operators for which a Wold-von Neumann decomposition type theorem has been established by Eschmeier and Langendörfer in [17]. Finally, in Section 5, we apply the results in Section 3 to obtain pure isometric dilation of certain important tuples of commuting contractions appearing in recent papers by Barik et al (see [7] and [6]).…”

“…Recently, the author proved that pair of commuting pure contractions with finite dimensional defect spaces admit pure isometric dilation (see [28]). The aim of this section is to apply Theorem 1.7 to extend the above result for certain important tuples of commuting pure contractions appearing in recent papers [7] and [6]. Let…”

“…It follows from a well-known computation that any element in SA n (E, E) is contractive. Let us begin by observing a result that follows from [Theorem 4.2 and Proposition 4.1, [7]]. For the sake of completeness we have included the proof of the result.…”

“…Let us begin by giving a brief introduction on the tuples of operators appearing in [7]. A particular class of these tuples were first studied by Grinshpan et al ([18]) in connection to multivariate von Neumann inequality.…”

On dilation and commuting liftings of n-tuples of commuting Hilbert space contractions

“…4,5]. On the other hand, the theory still remains of interest as indicated by the following list of recent results [2,3,4,6,7,9,13,14].…”