Isometric Dilations Research Articles

A Generalization of Ando’s Dilation, and Isometric Dilations for a Class of Tuples of q-Commuting Contractions

A Generalization of Ando’s Dilation, and Isometric Dilations for a Class of Tuples of q-Commuting Contractions

Isometric Dilations of Commuting Contractions and Brehmer Positivity

It is well-known that an n-tuple $$(n\ge 3)$$ 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).

High order isometric liftings and dilations

We show that a Hilbert space bounded linear operator has an $m$-isometric lifting for some integer $m\ge 1$ if and only if the norms of its powers grow polynomially. In analogy with unitary dilations of contractions, we prove that such operators also have an invertible $m$-isometric dilation. We also study $2$-isometric liftings of convex operators and $3$-isometric liftings of Foguel-Hankel operators.

Isometric dilations and von Neumann inequality for finite rank commuting contractions

Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by Pn(H), 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 Pn(H) 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 Cn.

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

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

Isometric dilations and 𝐻^{∞} calculus for bounded analytic semigroups and Ritt operators

We show that any bounded analytic semigroup on L p L^p (with 1 > p > ∞ 1>p>\infty ) whose negative generator admits a bounded H ∞ ( Σ θ ) H^{\infty }(\Sigma _\theta ) functional calculus for some θ ∈ ( 0 , π 2 ) \theta \in (0,\frac {\pi }{2}) can be dilated into a bounded analytic semigroup ( R t ) t ⩾ 0 (R_t)_{t\geqslant 0} on a bigger L p L^p -space in such a way that R t R_t is a positive contraction for any t ⩾ 0 t\geqslant 0 . We also establish a discrete analogue for Ritt operators and consider the case when L p L^p -spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier’s unitarization theorem.

Outgoing Cuntz scattering system for a coisometric lifting and transfer function

We study a coisometry that intertwines Popescu’s presentations of minimal isometric dilations of a given operator tuple and of a coisometric lifting of the tuple. Using this we develop an outgoing Cuntz scattering system which gives rise to an input–output formalism. A transfer function is introduced for the system. We also compare the transfer function and the characteristic function for the associated lifting.

Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations

AbstractIn this paper we propose a new technical tool for analyzing representations of Hilbert C*- product systems. Using this tool, we give a new proof that every doubly commuting representation over ℕk has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of .

Free holomorphic automorphisms of the unit ball of B(ℋ) n

In this paper we continue the study of free holomorphic functions on the noncommutative ball where B (ℋ) is the algebra of all bounded linear operators on a Hilbert space ℋ, and n = 1, 2, . . . or n = ∞. These are noncommutative multivariable analogues of the analytic functions on the open unit disc 𝔻 ≔ { z ∈ ℂ : | z | < 1}. The theory of characteristic functions for row contractions (elements in ) is used to determine the group of all free holomorphic automorphisms of [ B (ℋ) n ] 1 . It is shown that , the Moebius group of the open unit ball 𝔹 n ≔ { λ ∈ ℂ n : ∥ λ ∥ 2 < 1}. We show that the noncommutative Poisson transform commutes with the action of the automorphism group . This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra C *( S 1 , . . . , S n ), which leave invariant the noncommutative disc algebra . This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals new connections with noncommutative multivariable operator theory, especially, the theory of characteristic functions for row contractions and the noncommutative Poisson transforms. We show that the unitarily implemented automorphisms of the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra , respectively, are determined by the free holomorphic automorphisms of [ B (ℋ) n ] 1 , via the noncommutative Poisson transform. Moreover, we prove that . We also prove that any completely isometric automorphism of the noncommutative disc algebra has the form where . We deduce a similar result for the noncommutative Hardy algebra , which, due to Davidson-Pitts results on the automorphisms of , implies that is isomorphic to the group of contractive automorphisms of . We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group . This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under . Finally, in the last section, we prove a maximum principle for free holomorphic functions on the noncommutative ball [ B (ℋ) n ] 1 and provide some extensions of the classical Schwarz lemma to our noncommutative setting.

On isometric dilations of product systems of $C*$-correspondences and applications to families of contractions associated to higher-rank graphs

Let E be a product system of C*-correspondences over ℕ r 0 . Some sufficient conditions for the existence of a not necessarily regular isometric dilation of a completely contractive representation of E are established and difference between regular and * -regular dilations discussed. It is in particular shown that a minimal isometric dilation is *-regular if and only if it is doubly commuting. The case of product systems associated with higher-rank graphs is analysed in detail.

Elementary rotations of operators on regular Banach spaces

In this paper, we prove the existence of an elementary rotation (a Julia operator) for any continuous, linear, adjointable operator in a regular Banach space with inner product. The proof is based on a more general theorem of the author on the existence of an elementary rotation for any linear operator in a category with quadratic splitting. This result is a generalization of a well-known result about the existence of an elementary rotation for any continuous linear operator in a Krein space. The result can be useful for constructing isometric and unitary dilations as well as characteristic functions of continuous linear operators acting in regular Banach spaces with inner product.

ISOMETRIC DILATIONS OF REPRESENTATIONS OF PRODUCT SYSTEMS VIA COMMUTANTS

We construct a weak dilation of a not necessarily unital CP-semigroup to an E-semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E-semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to prove that covariant representations of product systems admit isometric dilations.

Hyperbolic geometry on the unit ball of $B(\mathcal{H})^{n}$ and dilation theory

In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball $[B(H)^n]_1^-$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, and its implications to noncommutative function theory. The central object is an intertwining operator $L_{B,A}$ of the minimal isometric dilations of $A, B\in [B(H)^n]_1^-$, which establishes a strong connection between noncommutative hyperbolic geometry on $[B(H)^n]_1^-$ and multivariable dilation theory. The goal of this paper is to study the operator $L_{B,A}$ and its connections to the hyperbolic metric $\delta$ on the Harnack parts of $[B(H)^n]_1^-$. We study the geometric structure of the operator $L_{B,A}$ and obtain new characterizations for the Harnack domination (resp. equivalence) in $[B(H)^n]_1^-$. We express $\|L_{B,A}\|$ in terms of the reconstruction operators $R_A$ and $R_B$, and obtain a Schwartz-Pick lemma for contractive free holomorphic functions on $[B(H)^n]_1$ with respect to the intertwining operator $L_{B,A}$. As a consequence, we deduce a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric.

Partially isometric dilations of noncommuting 𝑁-tuples of operators

Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

Unitary equivalence of operators and dilations

Given two contractions $T$ and $T'$ such that $T'-T$ is an operator of finite rank, we prove, under some conditions, the unitary equivalence of the unitary parts of the minimal isometric dilations (respectively minimal co-isometric extensions) of $T$ and&

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarason's characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Ptak defined the notion of generalized Hankel operator as follows: Given two contractions $$\mathcal{B}(\mathcal{H}_1 )$$ and $$\mathcal{B}(\mathcal{H}_2 )$$ , an operatorX: $$X:\mathcal{H}_1 \to \mathcal{H}_2 $$ is said to be a generalized Hankel operator ifT 2 X=XT 1 * andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT 1 andT 2. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Ptak's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.

Noncommutative joint dilations and free product operator algebras

Let An (n = 2, 3, . . . , or n = ∞) be the noncommutative disc algebra, and On (resp. Tn) be the Cuntz (resp. Toeplitz) algebra on n generators. Minimal joint isometric dilations for families of contractive sequences of operators on a Hilbert space are obtained and used to extend the von Neumann inequality and the commutant lifting theorem to our noncommutative setting. We show that the universal algebra generated by k contractive sequences of operators and the identity is the amalgamated free product operator algebra ∗CAni for some positive integers n1, n2, . . . , nk ≥ 1, and characterize the completely bounded representations of ∗CAni . It is also shown that ∗CAni is completely isometrically imbedded in the “biggest” free product C∗-algebra ∗CTni (resp. ∗COni), and that all these algebras are completely isometrically isomorphic to some universal free operator algebras, providing in this way some factorization theorems. We show that the free product disc algebra ∗CAni is not amenable and the set of all its characters is homeomorphic to (C1)1 × · · · × (Ck)1. An extension of the Naimark dilation theorem to free semigroups is considered. This is used to construct a large class of positive definite operator-valued kernels on the unital free semigroup on n generators and to study the class Cρ (ρ > 0) of ρ-contractive sequences of operators. The dilation theorems are also used to extend the operatorial trigonometric moment problem to the free product C∗-algebras ∗CTni and ∗COni .

On the Geometric Structure of Minimal Dilations on Hilbert $C*$-Modules

We build a unitary extension for an isometry on a Hilbert $C*-module and then, with this extension help, we obtain the minimal unitary dilation for an adjointable contraction starting from one of-its isometric dilations. Having as a starting point a result of B. Sz.-Nagy and C. Foias regarding the geometric structure of the minimal unitary dilations for Hubert space contractions we prove that this structure maintains itself on Hilbert modules. Finally, we present a necessary and sufficient condition on the minimal isometric dilation in order to admits a Wold-type decomposition, condition which also assures the complementability of the residual part space of the minimal unitary dilation.

Isometric dilations in nest algebras

LetT be a contraction acting in a separable Hilbert space\(\mathcal{H}\) and leaving invariant a nest\(\mathcal{N}\) of subspaces of\(\mathcal{H}\). We answer the question: when doesT have an isometric extension to\(\mathcal{H}\) ⊕\(\mathcal{H}\) which leaves invariant the nest\(\mathcal{N}\) ⊕\(\mathcal{N}\) = {N ⊕N :N ∈\(\mathcal{N}\);}.