Standard noncommuting and commuting dilations of commuting tuples

We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction, there are two commonly used dilations in multivariable operator theory. First there is the minimal isometric dilation consisting of isometries with orthogonal ranges, and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of the Cuntz algebra coming from dilations of commuting tuples.

     

Markov dilations on W1-algebras

In this paper a systematic study of Markov dilations is begun for completely positive operators on W¹-algebras which leave a faithful normal state invariant. It is shown that a minimal Markov dilation preserves important properties of the underlying completely positive operator. Afterwards some results are proved concerning the construction of dilations which lead to Markov dilations for large classes of operators. Finally some of the ideas developed here are applied to the study of a simple example over the 2 × 2 matrices.     

One class of C*-algebras generated by a family of partial isometries and multiplicators

We consider a C*-subalgebra of the algebra of all bounded operators on the Hilbert space of square-summable functions defined on some countable set. The algebra under consideration is generated by a family of partial isometries and the multiplier algebra isomorphic to the algebra of all bounded functions defined on the mentioned set. The partial isometry operators satisfy correlations defined by a prescribed map on the set. We show that the considered algebra is ?-graduated. After that we construct the conditional expectation from the latter onto the subalgebra responding to zero. Using this conditional expectation, we prove that the algebra under consideration is nuclear.     

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.     

Generalized inverses and similarity to partial isometries

We obtain some results related to the problems of Badea and Mbekhta (2005) [1] concerning the similarity to partial isometries using the generalized inverses. Especially, we involve the Moore-Penrose inverses. Also a characterization for such a similarity is given in the terms of dilations similar to unitary operators, which leads to a new criterion for the similarity to an isometry and to a quasinormal partial isometry.     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Criterion of Irreducibility

We study the operator algebra associated with a self-mapping ? on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l²(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.     

Semigroups of partial isometries

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.     

A Dilation Theory Approach to Cubature Formulas. II

An interpretation of multivariate cubature formulas for positive measures in terms of Hilbert space operators leads to a parametrization of all finite‐rank, cyclic and commutative dilations of a given cyclic tuple of self‐adjoint operators. Explicit matricial formulas for these dilations are presented; the abstract dilation problem suggested by the concrete computations is stated at the end of the note.     

Characteristic functions and joint invariant subspaces

Let T:=[T₁,…,Tn] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a “one-to-one” correspondence between the joint invariant subspaces under T₁,…,Tn, and the regular factorizations of the characteristic function ΘT associated with T. In particular, we prove that there is a non-trivial joint invariant subspace under the operators T₁,…,Tn, if and only if there is a non-trivial regular factorization of ΘT. We also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function, and prove the existence of joint invariant subspaces for certain classes of n-tuples of operators.We obtain criteria for joint similarity of n-tuples of operators to Cuntz row isometries. In particular, we prove that a completely non-coisometric row contraction T is jointly similar to a Cuntz row isometry if and only if the characteristic function of T is an invertible multi-analytic operator.     

Local isometries of

In this paper we study the structure of local isometries on . We show that when is first countable and is uniformly convex and the group of isometries of is algebraically reflexive, the range of a local isometry contains all compact operators. When is also uniformly smooth and the group of isometries of is algebraically reflexive, we show that a local isometry whose adjoint preserves extreme points is a -module map.

     

On Unitary Extensions of Multiplicative Families of Partial Isometries with a Generating Subspace

We consider a two parameter family of operators on a Hilbert space given by a multiplicative family of partial isometries with a generating subspace that commutes with an unitary group. The first parameter runs on a generalized interval of an abelian ordered group and the second on an abelian group. We show that it can be extended to a two parameter group of unitary operators on a larger Hilbert space. Applications to the problem of extending generalized semigroups of isometries and locally defined positive definite functions are given.     

On the symbol of nonlocal operators in Sobolev spaces

We consider nonlocal operators generated by pseudodifferential operators and the operator of shift along the trajectories of an arbitrary diffeomorphism of a smooth closed manifold. We introduce the notion of symbol of such operators acting in Sobolev spaces. As examples, we consider specific diffeomorphisms, namely, isometries and dilations.     

Multiplication Operators Between the Lipschitz Space and the Space of Bounded Functions on a Tree

In this paper, we characterize the bounded and the compact multiplication operators between the space of bounded functions on the set of vertices of a rooted infinite tree T and the Banach space of complex-valued Lipschitz functions on T. We also determine the operator norm and the essential norm for the bounded multiplication operators between these spaces and show that there are no isometries among such operators.     

Numerical Ranges of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">N</Emphasis>) Contractions

A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for C ₀(N) contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu.     

Nonnegative unitary operators

Unitary operators in Hilbert space map an orthonormal basis onto another. In this paper we study those that map an orthonormal basis onto itself. We show that a sequence of cardinal numbers is a complete set of unitary invariants for such an operator. We obtain a characterization of these operators in terms of their spectral properties. We show how much simpler the structure is in finite-dimensional space, and also describe the structure of certain isometries in Hilbert space.

     

Singular unitary dilations

We study the problem of determining which bounded linear operator on a Hilbert space can be dilated to a singular unitary operator. Some of the partial results we obtained are (1) every strict contraction has a diagonal unitary dilation, (2) everyC ₀ contraction has a singular unitary dilation, and (3) a contraction with one of its defect indices finite has a singular unitary dilation if and only if it is the direct sum of a singular unitary operator and aC ₀(N) contraction. Such results display a scenario which is in marked contrast to that of the classical case where we have the absolute continuity of the minimal unitary power dilation of any completely nonunitary contraction.     

Polar decomposition of oblique projections

The partial isometries and the positive semidefinite operators which appear as factors of polar decompositions of bounded linear idempotent operators in a Hilbert space are characterized.     

Dilations of operator-valued measures with bounded p-variations and framings on Banach spaces

The dilations for operator-valued measures (OVMs) and bounded linear maps indicate that the dilation theory is in general heavily dependent on the Banach space nature of the dilation spaces. This naturally led to many questions concerning special type of dilations. In particular it is not known whether ultraweakly continuous (normal) maps can be dilated to ultraweakly continuous homomorphisms. We answer this question affirmatively for the case when the domain algebra is an abelian von Neumann algebra. It is well known that completely bounded Hilbert space operator valued measures correspond to the existence of orthogonal projection-valued dilations in the sense of Naimark and Stinespring, and OVMs with bounded total variations are completely bounded but not the vice-versa. With the aim of classifying OVMs from the dilation point of view, we introduce the concept of total p-variations for OVMs. We prove that any completely bounded OVM has finite 2-variation, and any OVM with finite p-variation can be dilated to a (but usually non-Hilbertian) projection-valued measure of the same type. With the help of framing induced OVMs, we prove that conventional minimal dilation space of a non-trivial framing contains $c_0$, then does not have bounded p-variation.     

Joint Similarities and Parameterizations for Dilations of Dual g -frame Pairs in Hilbert Spaces

In this paper, we give an operator parameterization for the set of dilations of a given pair of dual g-frames and the set of dilations of pairs of dual g-frames of a given g-frame. In particular, for the dilations of a given pair of dual g-frames, we introduce the concept of joint complementary g-frames and prove that the joint complementary g-frames of a pair of dual g-frames are unique in the sense of joint similarity, which then helps to obtain a sufficient condition such that the complementary g-frames of a g-frame are unique in the sense of similarity and show that the set of dilations of a given dual g-frame pair are parameterized by a set of invertible diagonal operators. For the dilations of pairs of dual g-frames, we prove that the set of dilations of pairs of dual g-frames are parameterized by a set of invertible upper triangular operators.     

Hilbert空间中有界线性算子的膨胀

在Hilert空间中,有界线性算子的膨胀概念是算子的扩张概念的推广。它在讨论压缩算子的不变子空间时是有用的(见[1])。本文的目的是给出Hilbert空间上的有界线性算子B是它的闭子空间上的有界线性算子A的膨胀的一个充要条件。并且利用它给出恰当膨胀和拟恰当膨胀的概念、当讨论的是压缩算子的保范膨胀和酉膨胀时,它们是分别和极小保范膨胀和极小酉膨胀相当的。