John E. McCarthy, Orr Shalit
We show that whenever a contractive $k$-tuple $T$ on a finite dimensional space $H$ has a unitary dilation, then for any fixed degree $N$ there is a unitary $k$-tuple $U$ on a finite dimensional space so that $q(T) = P_H q(U) |_H$ for all polynomials $q$ of degree at most $N$.
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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. Firstly 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 rep…
The q-commuting contractions and $*$-regular dilations
本文引入q-交换压缩组的$*$-正则膨胀概念，推广了Gaspar-Suciu和Timotin关于交换压缩的结果。
Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant
本文通过数值实验表明，独立随机单位矩阵的通交扩张常数趋近于√2，进而证明任意收缩对的通交扩张常数严格小于2。
Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which 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 discuss how the basic properties of a dilation depend…
A contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the \wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe s…
We define four 3x3 commuting contractions which do not dilate to commuting isometries. However they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three $3\times3$ commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries.
Theory of $q$-commuting contractions-II: Regular dilation, Brehmer's positivity and von Neumann's inequality
本文研究q-交换收缩的正则扩张与布雷默正则性条件的关系，并建立冯·诺依曼型不等式。
We introduce the notion of $Q$-commuting operators which is a generalization of commuting operators. We prove a generalized version of commutant lifting theorem and Ando's dilation theorem in the context of $Q$-commuting operators.
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 $\mathbb{C}[z]$ or $\mathbb{C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of the isometric dilations, the von Neumann inequality and a sharper version of von …
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 [4].
Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \[ T_1T_2=QT_2T_1 \text{ or }T_1T_2=T_2QT_1 \text{ or }T_1T_2=T_2T_1Q. \] We give an explicit construction of isometric dilations for pairs of $Q$-commuting contractions for unitary $Q$, which generalizes the isometric dilation of Ando [2] for pairs of commuting contractions. In particular, for $Q=qI_{\mathcal{H}}$, where $q$ is a complex number of modulus $1$, this gives, as a corollary, an explicit construction of …
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.
A particular case of the fundamental Sz.-Nagy--Foias functional model for a contraction states that a pure contraction always dilates to a pure isometry. We are interested in the similar question for pairs, more precisely: does a pair of commuting pure contractions always dilate to a pair of commuting pure isometries? The purpose of this article is to identify pairs of commuting pure contractions for which the above question has an affirmative answer. Our method is based on an explicit structure of isometric dilation for pure pairs of commuting contractions obtained in a recent work by Das, Sa…
Non-commutative versions of Arveson's curvature invariant and Euler characteristic for a commuting $n$-tuple of operators are introduced. The non-commutative curvature invariant is sensitive enough to determine if an $n$-tuple is free. In general both invariants can be thought of as measuring the freeness or curvature of an $n$-tuple. The connection with dilation theory provides motivation and exhibits relationships between the invariants. A new class of examples is used to illustrate the differences encountered in the non-commutative setting and obtain information on the ranges of the invaria…
We show that every multivariable contractive weighted shift dilates to a tuple of commuting unitaries, and hence satisfies von Neumann's inequality. This answers a question of Lubin and Shields. We also exhibit a closely related $3$-tuple of commuting contractions, similar to Parrott's example, which does not dilate to a $3$-tuple of commuting unitaries.
In this paper we study extensions of commuting tuples of symmetric and isometric operators to commuting tuples of self-adjoint and unitary operators. Some conditions which ensure the existence of such extensions are presented. A multidimensional analog of the Godič-Lucenko Theorem is proved. An application to a multidimensional power-trigonometric moment problem is given.
On the universal commuting dilation constant
本文研究通用交换膨胀常数C_d，证明C_2 ≤ 2/√φ（φ为黄金分割比），将C_2的上下界收紧至1.5438 ≲ C_2 ≲ 1.5724，并改进了任意d元组的已知上下界。
This paper extends the trace formulas of [5] with perturbations in normed ideals of $B(H)$ to multivariate functions of commuting contractions admitting a dilation to commuting normal contractions.
In this paper, a new criterion for the similarity of commuting tuples of operators on Hilbert spaces is introduced. As an application, we obtain a geometric similarity invariant of tuples in the Cowen-Douglas class which gives a partial answer to a question raised by R.G. Douglas about the similarity of quasi-free Hilbert modules. Moreover, a new subclass of commuting tuples of Cowen-Douglas class is obtained.
We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Ando's dilation theorem for commuting contractions, Sz.-Nagy--Foias commutant lifting theorem, and Schur's representation for the unit ball of H^\infty, in the framework of noncommutative varieties in several variables and Poisson kernels on Fock spaces. This leads to inequalities which, in the particular case of two contractions, are sharper than Ando's inequality and Agler-McCarthy's inequality in the setting of commuting contractive matrices or arbitrary commuting contractions of class C_0…
Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz…
Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by $\mathcal{P}_n(\mathcal{H})$, of $n$-tuples of commuting contractions on a Hilbert space $\mathcal{H}$. We always assume that $n \geq 3$. 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, $n \geq 3$, of commuting contractions on Hilbert spaces (even in the level of finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, w…
Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has motivated lots of further works on model theory and generalizations to commuting tuples of $C_{\cdot0}$-contractions. In this paper, we focus on doubly commuting sequences of $C_{\cdot0}$-contractions, and establish the dilation theory and the analytic model theory for these sequences of operators. These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivaria…
In this paper, we study the relative perturbation bounds for joint eigenvalues of commuting tuples of normal $n \times n$ matrices. Some Hoffman-Wielandt type relative perturbation bounds are proved using the Clifford algebra technique. A result is also extended for diagonalizable matrices which improves a relative perturbation bound for single matrices.
One of the most important results in operator theory is Andô's \cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Schäffer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an Andô dilation for a commuting pair $(T_1,T_2)$ of contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of Andô's dilation theorem by giving…
Minimal unitary dilations for commuting contractions
本文研究交换压缩算子元组在乘积 $T$ 的极小等距与酉膨胀空间上的膨胀问题，给出伴随对称等价性、酉膨胀充要条件及 Schäffer、Sz. Nagy-Foias 与 $C._0$ 情形的显式构造。
An Ando-type dilation on right LCM monoids
本文提出了一种针对右LCM半群的Ando型扩张定理，将交换收缩和半群表示扩展为等距和等距表示，扩展了Barik和Das的先前结果。
Isometric dilation and Sarason's commutant lifting theorem in several variables
本文针对n≥3的可交换压缩算子元组，构造BCL型等距膨胀，证明von Neumann不等式及Sarason型交换子提升定理，并发现新的可膨胀算子类。
Classification and dilation for $q$-commuting $2 \times 2$ scalar matrices
本文对 $2\times 2$ 标量矩阵压缩的 $q$-交换算子组在相似意义下分为三类，并证明与这三类酉等价的算子组均具有 $\widetilde{q}$-酉膨胀，从而推广了 Holbrook 关于交换情形的经典膨胀结果。
Isometric, symmetric and isosymmetric commuting $d$-tuples of Banach space operators
该研究将Banach空间算子相关性质推广到交换$d$元组，阐明其诸多结构性质可通过单算子情形的常用论证推导，厘清该推广的内在逻辑。
We extend some of the results of Agler, Knese, and McCarthy [1] to $n$-tuples of commuting isometries for $n>2$. Let $\mathbb{V}=(V_1,\dots,V_n)$ be an $n$-tuple of a commuting isometries on a Hilbert space and let Ann$(\mathbb{V})$ denote the set of all $n$-variable polynomials $p$ such that $p(\mathbb{V})=0$. When Ann$(\mathbb{V})$ defines an affine algebraic variety of dimension 1 and $\mathbb{V}$ is completely non-unitary, we show that $\mathbb{V}$ decomposes as a direct sum of $n$-tuples $\mathbb{W}=(W_1,\dots,W_n)$ with the property that, for each $i=1,\dots,n$, $W_i$ is either a shift o…
Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples
本文基于Ando和Gurvits的工作，将Halmos定理推广到满足矩阵正性条件的Toeplitz压缩算子d元组，刻画了该条件，定义了新的距离度量，并证明了矩阵凸集包含的缩放常数存在。
Moment kernels, nested defects, and Cuntz dilations
本文研究随机算子元的矩核结构，证明其正性条件与Cuntz扩张的关系，并提出多级分解方法及曲率型不变量。
We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V…
In this document we study the local path connectivity of sets of $m$-tuples of commuting normal matrices with some additional geometric constraints in their joint spectra. In particular, given $\varepsilon>0$ and any fixed but arbitrary $m$-tuple $\mathbf{X}\in {M_n(\mathbb{C})}^m$ in the set of $m$-tuples of pairwise commuting normal matrix contractions, we prove the existence of paths between arbitrary $m$-tuples in the intersection of the previously mentioned sets of $m$-tuples in ${M_n(\mathbb{C})}^m$ and the $δ$-ball $B_ð(\mathbf{X},δ)$ centered at $\mathbf{X}$ for some $δ>0$, with respec…
On a minimal Andô dilation for a pair of strict contractions
本文改进安多扩张方法，构造了严格收缩在Hilbert空间上的最小等距扩张，并探讨了Banach空间和更一般规范空间中收缩对的安多扩张条件。
We explore aspects of dilation theory in the finite dimensional case and show that for a commuting $n$-tuple of operators $T=(T_1,...,T_n) $ acting on some finite dimensional Hilbert space $H$ and a compact set $X\subset \mathbb{C}^n$ the following are equivalent: 1. $T$ has a normal $ X$-dilation. 2. For any $m\in \mathbb{N}$ there exists some finite dimensional Hilbert space $K$ containing $H$ and a tuple of commuting normal operators $N=(N_1,...,N_n)$ acting on $K$ such that $$ q(T)=P_Hq(N)|_H$$ for all polynomials $q$ of degree at most $m$ and such that the joint spectrum of $N$ is contain…
We study analytic models of operators of class $C_{\cdot 0}$ with natural positivity assumptions. In particular, we prove that for an $m$-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space $\mathcal{H}$, there exists a Hilbert space $\mathcal{E}$ and a partially isometric multiplier $θ\in \mathcal{M}(H^2(\mathcal{E}), A^2_m(\mathcal{H}))$ such that \[\mathcal{H} \cong \mathcal{Q}_θ = A^2_m(\mathcal{H}) \ominus θH^2(\mathcal{E}), \quad \quad \mbox{and} \quad \quad T \cong P_{\mathcal{Q}_θ} M_z|_{\mathcal{Q}_θ},\]where $A^2_m$ is the weighted Bergman space and $H^2$ is the Hardy space over …
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.
Isometric Dilations for Representations of Product Systems
本文研究了乘积系统表示的等距扩张问题，证明在特定条件下完全收缩表示可扩张为等距表示，并展示了其在交换收缩元组中的应用。
We show that any m-isometric tuples of commuting operators on a finite dimensional Hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple. Our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables.
Rearrangement-invariant norms commuting with dilations
本文研究了具有排列不变性的空间中，范数与扩张算子相容的条件，证明了只有p-齐次范数满足此条件，并探讨了相关嵌入性质。
Weakly $\mathcal U(d)$-homogeneous commuting tuple of bounded operators
本文研究了弱 $\mathcal U(d)$-齐次交换元算子组，证明了多移位与 $\mathcal U(d)$-齐次多移位的等价性，并细化了球形平衡多移位的结果。
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…
Algebraic Andô Dilation
本文解决了向量空间上交换线性映射的 Andô 膨胀问题，证明了任意交换线性映射均可膨胀为交换的单射线性映射。
In this paper, we investigate the role of square functions defined for a $d$-tuple of commuting Ritt operators $(T_1,...,T_d)$ acting on a general Banach space $X$. Firstly, we prove that if the $d$-tuple admits a $H^\infty$ joint functional calculus, then it verifies various square function estimates. Then we study the converse when every $T_k$ is a $R$-Ritt operator. Under this last hypothesis, and when $X$ is a $K$-convex space, we show that square function estimates yield dilation of $(T_1,...,T_d)$ on some Bochner space $L_p(Ω;X)$ into a $d$-tuple of isomorphisms with a $C(\mathbb{T}^d)$ …
Dilation and Model Theory for Pairs of Commuting Contractions
本文将 Sz.-Nagy–Foias 膨胀与模型理论推广至压缩交换对，通过构造性证明、伪交换压缩提升及特征三元组，建立其完全酉不变量与函数模型。
Mackey Imprimitivity and commuting tuples of homogeneous normal operators
本文研究 Mackey 非本原性与齐次正规算子交换元组的关系，证明当紧空间分解为有限多个群轨道时，非本原性可相应直和分解，并建立与联合谱为轨道闭包的齐次正规元组的对应。
In this article, we show that a commuting pair $T=(T_1,T_2)$ of $\ast$-paranormal operators $T_1$ and $T_2$ with quasitriangular property satisfy the Weyl's theorem-I, that is $$σ_T(T)\setminusσ_{T_W}(T)=π_{00}(T)$$ and a commuting pair of paranormal operators satisfy Weyl's theorem-II, that is $$σ_T(T)\setminusω(T)=π_{00}(T),$$ where $σ_T(T),\, σ_{T_W}(T),\,ω(T)$ and $π_{00}(T)$ are the Taylor spectrum, the Taylor Weyl spectrum, the joint Weyl spectrum and the set consisting of isolated eigenvalues of $T$ with finite multiplicity, respectively. Moreover, we prove that Weyl's theorem-II holds …
For a commuting $d$- tuple of operators $\boldsymbol T$ defined on a complex separable Hilbert space $\mathcal H$, let $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ be the $d\times d$ block operator $\big (\!\!\big (\big [ T_j^* , T_i\big ]\big )\!\!\big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \b…
On simultaneous similarity of families of commuting operators
针对交换算子m元组同时相似性这一线性代数开放难题，提出任意Banach空间上k-循环交换算子m元组的相似性判别准则，推广了有限维下已有结果，并给出两类应用场景。
A commuting tuple of operators $(S_1,\dots, S_{n-1},P)$, defined on a Hilbert space $\mathcal H$, for which the closed symmetrized polydisc \[ Γ_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i \right): \,|z_i|\leq 1, i=1,\dots,n \right \} \] is a spectral set, is called a $Γ_n$-\textit{contraction}. A $Γ_n$-contraction is said to be \textit{pure} or $C._0$ if $P$ is $C._0$, that is, if ${P^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty$. We show that for any $Γ_n$-contraction $(S_1,\dots, S_{n-1},P)$, there is a unique operator tuple $(A_1…
For a tuple of operators $\boldsymbol{A}= (A_1, \ldots, A_d)$, $\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})$ is defined as $\min\limits_{\boldsymbol{z} \in \mathbb C^d} \|\boldsymbol{A-zI}\|$ and $\text{var}_x (\boldsymbol{A})$ as $\|\boldsymbol{A} x\|^2-\sum_{j=1}^d {\big|}\langle x| A_j x\rangle{\big|}^2.$ For a tuple $\boldsymbol{A}$ of commuting normal operators, it is known that $$\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})^2=\sup_{\|x\|=1}\text{var}_x (\boldsymbol{A}).$$ We give an expression for the maximal joint numerical range of a tuple of doubly commuting mat…
The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces $\mathcal{H}_K$ on the unit ball in $\mathbb C^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1}, \ldots ,M_{z_n})$ can be described in terms of suitable $\mathcal{H}_K$-inner functions. We prove that $\mathcal{H}_K$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way w…
On pure contractive semigroups
本文研究了 Hilbert 空间上的纯压缩半群，求出其换位子，并证明任意双重交换纯压缩半群元组可扩张为双重交换纯等距半群元组，进而得到等距半群元组的完整模型。
Let $\mathbb{V}$ denote an $n$-tuple of shifts of finite multiplicity, and denote by $\mathrm{Ann}(\mathbb{V})$ the ideal consisting of polynomials $p$ in $n$ complex variables such that $p(\mathbb{V})=0$. If $\mathbb{W}$ on $\mathfrak{K}$ is another $n$-tuple of shifts of finite multiplicity, and there is a $\mathbb{W}$-invariant subspace $\mathfrak{K}'$ of finite codimension in $\mathfrak{K}$ so that $\mathbb{W}|\mathfrak{K}'$ is similar to $\mathbb{V}$, then we write $\mathbb{V}\lesssim \mathbb{W}$. If $\mathbb{W}\lesssim \mathbb{V}$ as well, then we write $\mathbb{W}\approx \mathbb{V}$. In…
An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple $T$ can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H^2_{\mathcal{D}_{T^*}}(\mathbb{D}^n). As a consequence of the dilation theorem, we prove that there exists a closed subspace \mathcal{S}_T of the form \[\mathcal{…
We study interpolating sequences of $d$-tuples of matrices, by looking at the commuting and the non-commuting case separately. In both cases, we will give a characterization of such sequences in terms of separation conditions on suitable reproducing kernel Hilbert spaces, and we will give sufficient conditions stated in terms of separation via analytic functions. Examples of such interpolating sequences will also be given
Connecting $H^\infty$-functional calculus and isometric dilations for commuting families of Ritt$_E$ operators
本文证明，在UMD Banach空间上，Ritt$_E$算子交换族具有有界$H^\infty$-泛函演算当且仅当每个算子为$R$-Ritt$_E$且该族存在多项式有界的等距膨胀；在具有性质$(\alpha)$的空间中给出其他刻画。
Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. This extends a definition given by G. Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed.
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