arXiv math/05011542005-01-11math.FA

Perturbations of operators similar to contractions and the commutator equation

Catalin Badea

备注：: 26 pages, Preprint version
发表信息：: Studia Math. 150(2002), 273-293

英文摘要：

Let $T$ and $V$ be two Hilbert space contractions and let $X$ be a linear bounded operator. It was proved by C. Foias and J.P. Williams that in certain cases the operator block matrix $R(X;T,V)$ (defined in the text) is similar to a contraction if and only if the commutator equation $X = TZ-ZV$ has a bounded solution $Z$. We characterize here the similarity to contractions of some operator matrices $R(X;T,V)$ in terms of growth conditions or of perturbations of $R(0;T,V)=T\oplus V$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

大于恒等算子扰动的交换子

Commutators greater than a perturbation of the identity

本研究在有序赋范代数中证明：若正元$a$与元$b$的交换子大于恒等元的小扰动，则两元素范数乘积有对数型下界，同时给出希尔伯特格上满足该条件的正算子实例。

2023-11-09math.FA

多值Picard算子的容许扰动：Ćirić压缩条件；不动点与稳定性结果

Admissible perturbations of a multivalued Picard operator: Ćirić contraction condition; fixed point and stability results

本文研究了不满足Ćirić型压缩条件的多值算子的严格不动点与稳定性，通过分析Picard算子容许扰动$T_G$的条件，证明了结果依然成立，并以Takahashi意义下的扰动重新表述。

2025-10-30math.FA

p-Summable Commutators in Dimension d

We show that many invariant subspaces M for d-shifts (S_1,...,S_d) of finite rank have the property that the projection P onto M almost commutes with the S_k in the sense that the commutators PS_k - S_kP belong to the Schatten-von Neumann class L^p for every p > d. In such cases the d-tuple of operators (T_1,...,T_d) obtained by compressing (S_1,...,S_d) to the orthocomplement of M generates a *-algebra whose commutator ideal is contained in L^p, p > d. It follows that the C*-algebra generated by T_1,...,T_d is commutative modulo compact operators, the associated Dirac operator is Fredholm, an…

2003-08-11math.FAmath.OA

Trace formulas for tuples of commuting contractions

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.

2013-09-22math.FA

Dirac方程中长程扰动的初等交换子方法

Elementary commutator method for the Dirac equation with long-range perturbations

Graduate School of Mathematical Sciences, The University of Tokyo（东京大学大学院数理科学研究科）；Department of Mathematics, Graduate School of Science, Kobe University（神户大学理学研究科数学系）

本文提出了一种初等方法，用于研究Dirac方程在长程扰动下的广义特征函数不存在性和共振估计，为稳态散射理论提供理论基础。

2026-10-12math-phmath.APmath.FA

Functions of commuting contractions under perturbation

The purpose of the paper is to obtain estimates for differences of functions of two pairs of commuting contractions on Hilbert space. In particular, Lipschitz type estimates, Hölder type estimates, Schatten--von Neumann estimates are obtained. The results generalize earlier known results for functions of self-adjoint operators, normal operators, contractions and dissipative operators.

2018-04-06math.CAmath.CVmath.FA

更多相关论文收起更多论文

Bounded perturbations of Bernstein functions of several operator variables

The paper deals with (multidimensional and one-dimensional) Bochner-Phillips functional calculus. Bounded perturbations of Bernstein functions of (one or several commuting) semigroup generators on Banach spaces are considered, conditions for Lipschitzness and Frechet-differentiability of such functions are obtained, estimates for the norm of commutators are proved, and a generalization of Livschits-Kre\uın trace formula derived.

2016-11-20math.FA

A geometric approach to finite rank unitary perturbations

For a fixed natural number n, we consider a family of rank n unitary perturbations of a completely non-unitary contraction (cnu) with deficiency indices (n,n) on a separable Hilbert space. We relate the unitary dilation of such a contraction to its rank n unitary perturbations. Based on this construction, we prove that the spectra of the perturbed operators are purely singular if and only if the operator-valued characteristic function corresponding to the unperturbed operator is inner. In the case where n=1 the latter statement reduces to a well-known result in the theory of rank one perturbat…

2011-12-16math.FA

不是压缩相似的循环多项式有界算子的例子

Examples of cyclic polynomially bounded operators that are not similar to contractions

本文构造了不相似于压缩且拟相似于$C_0$-压缩或等距算子的循环多项式有界算子，基于Pisier构造的有限维算子序列的扰动，回答了Halmos问题。

2024-12-18math.FA

Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations. II

The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreĭn trace formula derived. The aim of the present paper is to prove the Frechet differe…

2018-02-03math.FA

Products of Block Toeplitz operators

In this paper we characterize when the product of two block Toeplitz operators is a compact perturbation of a block Toeplitz operator on the Hardy space of the open unit disk. Necessary and sufficient conditions are given for the commutator of two block Toeplitz operators to be compact.

1996-05-22math.FA

Examples of cyclic polynomially bounded operators that are not similar to contractions, II

The question if polynomially bounded operator is similar to a contraction was posed by Halmos and was answered in the negative by Pisier. His counterexample is an operator of infinite multiplicity, while all its restrictions on invariant subspaces of finite multiplicity are similar to contractions. In [G16], cyclic polynomially bounded operators which are not similar to contractions was constructed. The construction was based on a perturbation of the sequence of finite dimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, construct…

2018-05-15math.FA

General Clark model for finite rank perturbations

All unitary perturbations of a given unitary operator $U$ by finite rank $d$ operators with fixed range can be parametrized by $(d\times d)$ unitary matrices $Γ$; this generalizes unitary rank one ($d=1$) perturbations, where the Aleksandrov--Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\subset\mathbb{C}$. For a purely contractive $Γ$ the resulting perturbed operator $T_Γ$ is a contraction (a completely non-unitary contraction under the natural assumption about cyclicity of the range), so they admit the functional model. In this paper we in…

2017-10-30math.FA

Perturbation bounds for eigenspaces under a relative gap condition

A basic problem in operator theory is to estimate how a small perturbation effects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, taylored for structured random perturbations. As a main example, we consider the empirical covariance operator, and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches.

2018-12-16math.FAmath.PR

Contracting dynamical systems in Banach spaces

Contraction rates of time-varying maps induced by dynamical systems illuminate a wide range of asymptotic properties with applications in stability analysis and control theory. In finite-dimensional smoothly varying inner-product spaces such as $\mathbb{R}^n$ and $\mathbb{C}^n$ with Riemannian metrics, contraction rates can be estimated by upper-bounding the real numerical range of the vector field's Jacobian. However, vector spaces with norms other than $L^2$ commonly arise in the stability analysis of infinite-dimensional systems such as those arising from partial differential equations and …

2022-01-08math.DSmath.FA

The behavior of functions of operators under perturbations

This is a survey article. We consider different problems in connection with the behavior of functions of operators under perturbations of operators. We deal with three classes of operators: unitary operators, self-adjoint operators, and contractions. We study operator Lipschitz and operator differentiable functions. We also study the behavior of functions under perturbations of an operator by an operator of Schatten--von Neumann class $\bS_p$ and apply the results to the Livschits--Krein and Koplienko--Neidhardt trace formulae. We also include in this survey article recent unexpected results o…

2009-04-10math.CAmath.CVmath.FA

Perturbations of completely positive maps and strong NF algebras

Let $ϕ:M_n\to B(H)$ be an injective, completely positive contraction with $\Vϕ^{-1}:ϕ(M_n)\to M_n\V_{cb}\leq1+δ(ε).$ We show that if either (i) $ϕ(M_n)$ is faithful modulo the compact operators or (ii) $ϕ(M_n)$ approximately contains a rank 1 projection, then there is a complete order embedding $ψ:M_n\to B(H)$ with $\Vϕ-ψ\V_{cb}<ε.$ We also give examples showing that such a perturbation does not exist in general. As an application, we show that every $C^*$-algebra $A$ with $\mathcal{OL}_\infty(A)=1$ and a finite separating family of primitive ideals is a strong NF algebra, providing a partial …

2009-06-18math.FAmath.OA

Functions of noncommuting operators under perturbation of class $\boldsymbol{S}_p$

In this article we prove that for $p>2$, there exist pairs of self-adjoint operators $(A_1,B_1)$ and $(A_2,B_2)$ and a function $f$ on the real line in the homogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$ such that the differences $A_2-A_1$ and $B_2-B_1$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ but $f(A_2,B_2)-f(A_1,B_1)\not\in\boldsymbol{S}_p$. A similar result holds for functions of contractions. We also obtain an analog of this result in the case of triples of self-adjoint operators for any $p\ge1$

2019-02-17math.CAmath.CVmath.FA