Isometric Dilation Research Articles

Unital positive Schur multipliers on $S_n^p$ with a completely isometric dilation

Let $1<p\neq 2<\infty $ and let $S^p_n$ be the associated Schatten von Neumann class over $n\times n$ matrices. We prove new characterizations of unital positive Schur multipliers $S^p_n\to S^p_n$ which can be dilated into an invertible complete isometry acting on a non-commutative $L^p$-space. Then we investigate the infinite dimensional case.

Dilation properties of measurable Schur multipliers and Fourier multipliers

In the article, we find new dilatation results on non-commutative \(L^p\) spaces. We prove that any self-adjoint, unital, positive measurable Schur multiplier on some \(B(L^2(\Sigma ))\) admits, for all \(1\leqslant p<\infty \), an invertible isometric dilation on some non-commutative \(L^p\)-space. We obtain a similar result for self-adjoint, unital, completely positive Fourier multiplier on VN(G), when G is a unimodular locally compact group. Furthermore, we establish multivariable versions of these results.

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.

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

Abstract The n-tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an n-tuple. A series of such liftings leads to an isometric dilation of the n-tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive definite n-tuple has an isometric dilation.

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.

The Reducibility of Truncated Toeplitz Operators

Let T be a contraction on the Hilbert space $$\mathscr {H}$$ and S a minimal isometric dilation of T. In this paper, we show that every projection in $$\{T\}'$$ can be extended to a projection in $$\{S\}'$$ . Using this result, a sufficient condition for reducibility of $$A^{\theta }_{B_{n}}$$ , where $$B_{n}$$ is a finite Blaschke product with order n, is given. In particular, we determine when $$A^{\theta }_{B_{n}}$$ is reducible in two special cases. One case is that $$n=2,3$$ and the other case is that $$B_{n}=z^{n}$$ ( $$n\in \mathbb {N}$$ ) and $$\theta $$ is a singular inner function.

Classes of operators related to m-isometric operators

Isometries played a pivotal role in the development of operator theory, in particular with the theory of contractions and polar decompositions and has been widely studied due to its fundamental importance in the theory of stochastic processes, the intrinsic problem of modeling the general contractive operator via its isometric dilation and many other areas in applied mathematics. In this paper we present some properties of n-quasi-(m;C)-isometric operators. We show that a power of a n-quasi-(m;C)-isometric operator is again a n-quasi-(m;C)-isometric operator and some products and tens

Regular Dilation and Nica-covariant Representation on Right LCM Semigroups

Regular dilation has recently been extended to graph product of $$\mathbb {N}$$ , where having a $$*$$ -regular dilation is equivalent to having a minimal isometric Nica-covariant dilation. In this paper, we first extend the result to right LCM semigroups, and establish a similar equivalence among $$*$$ -regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result can be applied to various semigroups to establish conditions for $$*$$ -regular dilation.

Regular dilation on graph products of [formula omitted

We extended the definition of regular dilation to graph products of N, which is an important class of quasi-lattice ordered semigroups. Two important results in dilation theory are unified under our result: namely, Brehmer's regular dilation on Nk and Frazho–Bunce–Popescu's dilation of row contractions. We further show that a representation of a graph product has an isometric Nica-covariant dilation if and only if it is ⁎-regular. A special case of our result was considered by Popescu, and we studied the connection with Popescu's work.

Regular representations of lattice ordered semigroups

We establish a necessary and sufficient condition for a representation of a lattice ordered semigroup to be regular, in the sense that certain extensions are completely positive definite. This result generalizes a theorem due to Brehmer where the lattice ordered group was taken to be $\mathbb{Z}_+^\Omega$. As an immediate consequence, we prove that contractive Nica-covariant representations on lattice ordered semigroups are regular, and therefore, its minimal isometric dilation is also Nica-covariant. We also introduce an analog of commuting row contractions on lattice ordered group and show that such a representation is regular.

Standard commuting dilations and liftings

We identify how the standard commuting dilation of the maximal commuting piece of any row contraction, especially on a finite dimensional Hilbert space, is associated to the minimal isometric dilation of the row contraction. Using the concept of standard commuting dilation it is also shown that if liftings of row contractions are on finite dimensional Hilbert spaces, then there are strong restrictions on properties of liftings

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 .

Maximum A-isometric part of an A-contraction and applications

For two bounded linear operators A and T on a complex Hilbert space H (A being positive) which satisfy the inequality T*AT ≤ A, we study the maximum subspace ℳ0 which reduces A and T, on which the equality T*AT = A holds. We show that in some cases involving the condition AT = A1/2TA1/2, ℳ0 can be expressed in terms of the minimal isometric dilation of the contraction \( \hat T \) on \( \overline {R(A)} \) associated to T by the condition \( \hat T \)A1/2 = A1/2T. As application we find a concrete representation for ℳ0 when T is a contraction with ST = ST2, where ST is the strong limit of the sequence [T*nTn: n ≥ 1]. Also, we derive some applications for hyponormal contractions and quasi-isometries.

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.

Subproduct systems

The notion of a subproduct system, a generalization of that of a product system, is introduced. We show that there is an essentially 1 to 1 correspondence between cp -semigroups and pairs (X,T) where X is a subproduct system and T is an injective subproduct system representation. A similar statement holds for subproduct systems and units of subproduct systems. This correspondence is used as a framework for developing a dilation theory for cp -semigroups. Results we obtain: (i) a *-automorphic dilation to semigroups of *-endomorphisms over quite general semigroups; (ii) necessary and sufficient conditions for a semigroup of CP maps to have a *-endomorphic dilation; (iii) an analogue of Parrot's example of three contractions with no isometric dilation, that is, an example of three commuting, contractive normal CP maps on B(H) that admit no *-endomorphic dilation (thereby solving an open problem raised by Bhat in 1998). Special attention is given to subproduct systems over the semigroup {N} , which are used as a framework for studying tuples of operators satisfying homogeneous polynomial relations, and the operator algebras they generate. As applications we obtain a noncommutative (projective) Nullstellensatz, a model for tuples of operators subject to homogeneous polynomial relations, a complete description of all representations of Matsumoto's subshift C^*-algebra when the subshift is of finite type, and a classification of certain operator algebras – including an interesting non-selfadjoint generalization of the noncommutative tori.

Lifting fixed points of completely positive mappings

Let 1 ⩽ n ⩽ ∞ , and let { T j } j = 1 n ⊂ B ( H ) be a row contraction on some Hilbert space H . Let F ( T ) be the space of all X ∈ B ( H ) such that ∑ j = 1 n T j X T j ∗ = X . We show that, if non-zero, this space is completely isometric to the commutant of the Cuntz part of the minimal isometric dilation of { T j } j = 1 n .

Formula omitted]-dilation of strongly commuting [formula omitted]-semigroups

We prove that every strongly commuting pair of CP 0 -semigroups has a minimal E 0 -dilation. This is achieved in two major steps, interesting in themselves: (1) we show that a strongly commuting pair of CP 0 -semigroups can be represented via a two parameter product system representation; (2) we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP 0 -semigroups on B ( H ) , H finite-dimensional, has an E 0 -dilation.

REGULAR DILATIONS OF REPRESENTATIONS OF PRODUCT SYSTEMS

We study completely contractive representations of product systems X of correspondences over the semigroup Z+. We present a necessary and sufficient condition for such a representation to have a regular isometric dilation. We discuss representations that doubly commute and show that these representations induce completely contractive representations of the norm closed algebra generated by the image of the Fock representation of X.

Minimal Cuntz-Krieger dilations and representations of Cuntz-Krieger algebras

Given a contractive tuple of Hilbert space operators satisfying certainA-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximalA-relation piece. We define a maximal piece more generally for a finite set of polynomials inn noncommuting variables. We classify all representations of Cuntz-Krieger algebrasO A obtained from dilations of commuting tuples satisfyingA-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.