Dmitriy S. Kalyuzhniy
The explicit constructions of minimal isometric, and minimal unitary dilations of an arbitrary linear pencil of operators $T(λ)=T_0+λT_1$ consisting of contractions on a separable Hilbert space for $|λ|=1$, which generalize the classical constructions (the case $T_1=0$), are presented. In contrast to the classical case these dilations are essentially non-unique.
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We provide an explicit normal distinguished boundary dilation to a pair of commuting operators $(S,P)$ having the closed symmetrized bidisc $Γ$ as a spectral set. This is called Sz.-Nagy dilation of $(S,P)$. The operator pair that dilates $(S,P)$ is obtained by an application of Stinespring dilation of $(S,P)$ given by Agler and Young. We further prove that the dilation is minimal and the dilation space is no bigger than the dilation space of the minimal unitary dilation of the contraction $P$. We also describe model space and model operators for such a pair $(S,P)$.
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…
Inspired by the Sz.-Nagy-Foias dilation theorem we show that $n$ freely independent contractions dilate to $n$ freely independent unitaries.
The principal theorem of Sz.-Nagy on dilation of a positive definite Hilbert space operator valued function has played a central role in the development of the non-self-adjoint operator theory. In this paper we introduce the positive definiteness for locally Hilbert space operator valued kernels, we prove an analogue of the Sz.-Nagy dilation theorem and, as application, we obtain dilation results for locally contractions and locally $ρ$ - contractions as well as for locally semi-spectral measures.
Dilation and Model Theory for Pairs of Contractions with a Twisted Commutation Relation
本文针对幺模复数q对应的q-交换压缩算子对，发展了与单压缩算子经典Sz.-Nagy--Foias膨胀和模型论平行的理论。
We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if $X$ is a super-reflexive Banach space and $T$ is contained in the weakly closed convex hull of all invertible isometries on $X$, then $T$ admits a dilation to an invertible isometry on a Banach space $Y$ with the same regularity as $X$. The classical dilation theorems of Sz.-Nagy and Akcoglu-Sucheston are easy consequences of our general theory.
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…
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。
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…
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 …
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…
A tuple of commuting Hilbert space operators $(S_1, \dots, S_{n-1}, P)$ having the closed symmetrized polydisc \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n} z_iz_j, \cdots, \prod_{i=1}^{n}z_i\right) : |z_i|\leq 1\,, \; \; \; 1\leq i \leq n-1 \right\} \] as a spectral set is called a $Γ_n$-contraction. From the literature we have that a point $(s_1, \dots , s_{n-1},p)$ in $Γ_n$ can be represented as $s_i=c_i+pc_{n-i}$ for some $(c_1, \dots, c_{n-1}) \in Γ_{n-1}$. We construct a minimal $Γ_n$-isometric dilation for a particular class of c.n.u. $Γ_n$-contractions $(S_1, …
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。
Functional models for $Γ_n$-contractions
本文提出了一种针对Γ_n-收缩的函数模型，基于Douglas模型和分解结果，得到了Sz.-Nagy-Foias型和Schäffer型函数模型。
A classical result of Sz.-Nagy asserts that a Hilbert-space contraction operator $T$ can be lifted to an isometry $V$. A more general multivariable setting of recent interest for these ideas is the case where (i) the unit disk is replaced by a certain domain contained in ${\mathbb C}^3$ (called the {\em tetrablock}), (ii) the contraction operator $T$ is replaced by a commutative triple $(T_1, T_2, T)$ of Hilbert-space operators having ${\mathbb E}$ as a spectral set (a tetrablock contraction) . The rational dilation question for this setting is whether a tetrablock contraction $(T_1, T_2, T)$ …
Minimal isometric dilations and operator models for the polydisc
针对希尔伯特空间上满足特定条件的交换压缩算子，研究其扩张为交换等距算子的充要条件，构造两类等距膨胀及特殊情形下的不同膨胀，由此得到基于向量Hardy空间乘法算子的算子模型，还推导了条件分解并给出例子。
Lamperti Operators, Dilation Theory, and Applications in Noncommutative Ergodic Theory
本文提出非交换环境下定量遍历定理的新框架，结合Lamperti运算符和扩张理论，扩展了经典定理并应用于非交换 $L_p$-空间。
In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectrum of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.
A new functional model for contractions
本文利用对称算子扩张理论的共同几何基础，为完全非酉压缩算子提出一种内蕴、规范构造且不依赖特征函数的新泛函模型。
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor $\vartheta(d)$, of all d-by-d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. An analytic formula for $\vartheta(d)$ is derived, which as a by-product gives new probabilistic results for the binomial and beta distributions. Dilating to commuting operators has consequences…
Spectral set, complete spectral set and dilation for Banach space operators
本文通过反例证明Hilbert空间中压缩算子、谱集、等距膨胀与完全谱集之间的经典等价关系在Banach空间算子中均不成立，并利用Bohr定理刻画了谱集与Bohr半径的关系，给出Hilbert空间的若干新刻画。
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…
We suggest a new version of the notion of $ρ$-dilation ($ρ>0$) of an $N$-tuple $\mathbf{A}=(A_1,...,A_N)$ of bounded linear operators on a common Hilbert space. We say that $\mathbf{A}$ belongs to the class $C_{ρ,N}$ if $\mathbf{A}$ admits a $ρ$-dilation $\widetilde{\mathbf{A}}=(\widetilde{A}_1,...,\widetilde{A}_N)$ for which $ζ\widetilde{\mathbf{A}}:=ζ_1\widetilde{A}_1+... +ζ_N\widetilde{A}_N$ is a unitary operator for each $ζ:=(ζ_1,...,ζ_N)$ in the unit torus $\mathbb{T}^N$. For N=1 this class coincides with the class $C_ρ$ of B. Sz.-Nagy and C. Foiaş. We generalize the known descriptions of…
A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain $Ω$ contained in ${\mathbb C}^d$, (ii) the contraction operator $T$ is replaced by a commuting tuple $\bfT = (T_1, \dots, T_d)$ such that $\| r(T_1, \dots, T_d) \|_{\cL(\cH)} \le \sup_{\lam \in Ω} | r(\lam) |$ for all rational functions with no singularities in $\overlineΩ$ and the unitary operator $\cU$ is replaced by an $Ω$-unitary operator tuple, i.e., a commut…
Necessary Conditions for $Γ_{E(3; 3; 1, 1, 1)}$-Isometric Dilation, $Γ_{E(3; 2; 1, 2)}$-Isometric Dilation and $\mathcal{\bar{P}}$-Isometric Dilation
本文证明了$Γ_{E(3; 3; 1, 1, 1)}$、$Γ_{E(3; 2; 1, 2)}$与$\mathcal{\bar{P}}$-等距膨胀存在的必要条件，构造了特定反例表明现有充分条件非必要，并给出了显式等距膨胀构造。
The spectral and scattering properties of non-selfadjoint problems pose a mathematical challenge. Apart from exceptional cases, the well-developed methods used to examine the spectrum of selfadjoint problems are not applicable. One of the tools to attack non-selfadjoint problems are functional models. A drawback of many functional models is that their constructions require objects which may be difficult to describe explicitly, such as operator square roots, making it hard to apply the results to specific examples. We develop a functional model for the case when the non-selfadjointness arises b…
A commuting tuple of $n$ operators $(S_1, \dots, S_{n-1}, P)$ defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n}z_iz_j, \dots, \prod_{i=1}^{n}z_i \right) : |z_i|\leq 1, i=1, \dots, n \right\} \] is a spectral set is called a $Γ_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called an $\mathbb E$-contraction, where \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3 \neq 0 \quad \fora…
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