Wold decomposition for representations of product systems of C*-correspondences

图上自相似半群作用的 Wold 分解与 C*-包络

Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs

本文建立图上自相似半群作用表示的 Wold 分解，并利用显式非平凡膨胀证明 $\mathcal{A}_{\mathbb{N},E}$ 的 C*-包络就是 $\mathcal{O}_{\mathbb{Z},E}$。

Axioms required to get Wold-Type decomposition

Three canonical decompositions concerning commuting pair of isometries, power partial isometries, and contractions are reassessed. They have already been proved in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. Ignoring topological structures, we give an algebraic approach to obtain them in the larger category of Baer $*$-rings.

Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of op…

Doubly commuting invariant subspaces for representations of product systems of $C^*$-correspondences

We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariant representation of a product system of $C^*$-correspondences. This extends a recent Wold-type decomposition by Jeu and Pinto for a $q$-doubly commuting isometries. Application to the wandering subspaces of doubly commuting induced representations is explored, and a version of Mandrekar's Beurling type theorem is obtained to study doubly commuting invariant subspaces using Fock space approach due to Popescu.

协变表示的Wold型分解与Beurling型定理

Wold-type decomposition and Beurling-Type Theorem for Covariant Representations

利用算子不等式研究协变表示的Wold型分解，证明每个非零不变子空间由其游荡子空间唯一确定，将Beurling定理推广到C*-对应的协变表示。

具有相等值域的等距算子的 Wold 分解

Wold decomposition for isometries with equal range

本文研究具有相等值域的等距算子 $n$ 元组，证明其存在唯一 Wold 分解，建立解析模型，并表明游走数据构成完备酉不变量，统一了已有结果。

关于扭曲压缩算子对的分解

On decomposition for pairs of twisted contractions

本文研究希尔伯特空间上扭曲压缩算子对的Wold型分解，推导双扭曲等距算子对的分解结论，给出Słoćinski定理的新证明，得到显式分解结果与极小等距膨胀空间上的唯一双扭曲等距对，证明等距-余等距扭曲对的双扭曲性并给出双扭曲等距对的刻画。

C^*-algebras generated by scaling elements

We investigate C^*-algebras generated by scaling elements. We generalize the Wold decomposition and Coburn's theorem on isometries to scaling elements. We also completely determine when the C^*-algebra generated by a scaling element contains an infinite projection.

Generating wandering subspaces for doubly commuting covariant representations

We obtain a Halmos-Richter-type wandering subspace theorem for covariant representations of C*-correspondences. Further the notion of Cauchy dual and a version of Shimorin's Wold-type decomposition for covariant representations of C*-correspondences is explored and as an application a wandering subspace theorem for doubly commuting covariant representations is derived. Using this wandering subspace theorem generating wandering subspaces are characterized for covariant representations of product systems in terms of the doubly commutativity condition.

A caricature of dilation theory

We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations, BCL theorem etc. We point out some natural generalizations and variations.

The structure of an isometric tuple

An $n$-tuple of operators $(V_1,...,V_n)$ acting on a Hilbert space $H$ is said to be isometric if the operator $[V_1\...\ V_n]:H^n\to H$ is an isometry. We prove a decomposition for an isometric tuple of operators that generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

Orthogonal decompositions and twisted isometries II

We classify tuples of (not necessarily commuting) isometries that admit von Neumann-Wold decomposition. We introduce the notion of twisted isometries for tuples of isometries and prove the existence of orthogonal decomposition for such tuples. The former classification is partially inspired by a result that was observed more than three decades ago by Gaspar and Suciu. And the latter result generalizes Popovici's orthogonal decompositions for pairs of commuting isometries to general tuples of twisted isometries which also includes the case of tuples of commuting isometries. Our results unify al…

Doubly $Λ$-commuting row isometries, universal models, and classification

The goal of the paper is to study the structure of the k-tuples of doubly $Λ$-commuting row isometries and the $C^*$-algebras they generate from the point of view of noncommutative multivariable operator theory. We obtain Wold decompositions, in this setting, and use them to classify the $k$-tuples of doubly $Λ$-commuting row isometries up to a unitary equivalence. We introduce a universal model in this setting, describe its invariant subspaces, and develop a dilation theory on $Λ$-polyballs.

等距算子的双交换半群

Doubly Commuting Semigroups of Isometries

本文研究等距算子的双交换半群结构，给出希尔伯特模下Cooper定理的新证明，分析有限d时$\u211d_{+}^{d}$对应Fell拓扑的$T_0$性质，同时揭示d为无穷时的多种拓扑与分解反常现象。

环带中算子的联合约化子空间与正交分解

Joint reducing subspaces and orthogonal decompositions of operators in an annulus

本文研究环带谱集对应的希尔伯特空间交换算子组，刻画$\mathbb A_r^n$-酉与等距算子结构，得到其Wold型分解并推广至压缩算子族。

$\mathbb{N}_0^{k}$上积系统等距表示的不变子空间刻画

A characterization of invariant subspaces for isometric representations of product system over $\mathbb{N}_0^{k}$

本文借助 Wold-von Neumann 分解刻画 $\mathbb{N}_0^{k}$ 上积系统双交换纯等距表示的换位子与不变子空间，给出同构不变量并分类一大类交换等距表示。