Classes of Operators Similar to Partial Isometries

The present paper deals with operators similar to partial isometries. We get some (necessary and) sufficient conditions for the similarity to (adjoints of) quasinormal partial isometries, or more general, to power partial isometries. We illustrate our results on the class of n-quasi-isometries, obtaining that a n-quasi-isometry is similar to a power partial isometry if and only if the ranges are closed. In particular if n = 2, these conditions ensure the similarity to quasinormal partial isometries of Duggal and Aluthge transforms of 2-quasi-isometries. The case when a n-quasi-isometry is a partial isometry is also studied, and a structure theorem for n-quasi-isometries which are power partial isometries is given. The second author was partially supported by Romanian 2-CEX Research grant. no. 06-11-34/2006.     

初等算子的正规性

设初等算子E（X）＝n↑∑↑i=1AiXBi，定义E^*(X)=n↑∑↑i=1Ai^*XBi^*我们证明了EE^*＝E^*E当县仅当｛Ai｝和｛Bi｝都是交换的正规算子族，从而回答了由D.Keckic提出的关于初等算子正规性的开问题。我们还给出了E＝E^*的充分必要条件。     

Semigroups of Partial Isometries and Symmetric Operators

Let \({{\{ V(t) | \ t \in [0 , \infty) \}}}\) be a one-parameter strongly continuous semigroup of contractions on a separable Hilbert space and let V(?t) : = V*(t) for \({t \in [0, \infty)}\). It is shown that if V(t) is a partial isometry for all \({t \in [-t_0 , t_0], t_0 > 0}\), then the pointwise two-sided derivative of V(t) exists on a dense domain of vectors. This derivative B is necessarily a densely defined symmetric operator. This result can be viewed as a generalization of Stone’s theorem for one-parameter strongly continuous unitary groups, and is used to establish sufficient conditions for a self-adjoint operator on a Hilbert space \({\mathcal{K}}\) to have a symmetric restriction to a dense linear manifold of a closed subspace \({\mathcal H \subset \mathcal K}\). A large class of examples of such semigroups consisting of the compression of the unitary group generated by the operator of multiplication by the independent variable in \({\mathcal {K} := \oplus _{i=1} ^n L^2 (\mathbb {R})}\) to certain model subspaces of the Hardy space of n?compenent vector valued functions which are analytic in the upper half plane is presented.     

Essential Normality of Homogeneous Submodules

Let M ì H(\mathbbB){M \subset H(\mathbb{B})} be a homogeneous submodule of the n-shift Hilbert module on the unit ball in \mathbbCⁿ{\mathbb{C}^{n}}. We show that a modification of an operator inequality used by Guo and Wang in the case of principal submodules is equivalent to the existence of factorizations of the form [M_zj^*,P_M] = (N+1)^-1A_j{[M_{z_j}^*,P_M] = (N+1)^{-1}A_j}, where N is the number operator on H(\mathbbB){H(\mathbb{B})}. Thus a proof of the inequality would yield positive answers to conjectures of Arveson and Douglas concerning the essential normality of homogeneous submodules of H(\mathbbB){H(\mathbb{B})}. We show that in all cases in which the conjectures have been established the inequality holds and leads to a unified proof of stronger results.     

Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators

The Keldysh theorem is generalized to an arbitrary closed operator that is not necessarily close to self-adjoint operators and has a resolvent of Schatten–von Neumann class S _p . Based on this theorem, conditions of spectrum localization are obtained for certain classes of non-self-adjoint differential operators.     

广义弱亚正规算子的正规性

主要研究了T是广义弱亚正规算子时,T~t是拟正规算子的充要条件是T是拟正规算子.并且举例说明了存在非次正规的广义弱亚正规算子T使得T~t是次正规的.     

Dual Toeplitz Operators on the Sphere Via Spherical Isometries

We solve a characterization problem for dual Hardy-space Toeplitz operators on the unit sphere \({\mathbb{S}_{n}}\) in \({\mathbb{C}^{n}}\) posed by Guediri (Acta Math Sin (English series) 29(9):1791–1808, 2013). Our proof relies on the observation that dual Toeplitz operators on the orthogonal complement \({H^{2}(\mathbb{S}_{n})^{\bot}}\) of the Hardy space in L ² can be viewed as Toeplitz operators with respect to a suitable spherical isometry. This correspondence also allows us to determine the commutator ideal of the dual Toeplitz C ^*-algebra.     

算子的本性相似

对作用于一复可分无穷维Hilbet空间(？)上的两个有界线性算子T,S,它 们本性相似和π(T),π(S)(T,S在Calkin代数中的象)相似是否等价？这是有趣的问 题.本文引入一个算子类ES((？)),并证明了对ES((？))中的算子,上面提到的两种相 似性是等价的. 此外,还证明了ES((？))在B((?))中按照算子范数拓扑稠密.     

一类拟相似算子的本质谱

引进亚控制算子的概念,通过算子谱的精密结构的分析,给出拟相似亚控制算子的本质谱相等的若干条件,所得结果改进和发展了Williams L R的有关结果.     

Essential Norms of Composition Operators

We obtain simple estimates for the essential norm of a composition operator acting from the Hardy space H ^p to H ^q , p > q, in one or several variables. When p = and q = 2 our results give an exact formula for the essential norm.     

On the Normality of the Sum of Two Normal Operators

The aim of this paper is to give sufficient conditions on two normal operators (bounded or not), defined on a Hilbert space, which make their algebraic sum normal. The results are accompanied by some interesting examples and counter examples.     

Semigroups Generated by Similarity Orbits of Semi-invertible Operators and Unitary Orbits of Isometries

It is proved that for a left invertible operator A, which is not a sum of a scalar and a compact operator, the semigroup generated by similarity orbit of A contains all left invertible operators with suitable (large enough) deficiency. A corresponding result holds for right invertible operators. For isometries, the same statement is proved for unitary orbits.     

算子的左本质谱与Browder本质谱

设算子S和T拟相似,应用算子谱的精密结构的分析,证明了Browder本质谱σB(S)的连通分支与σB(T)的某些子集的相交关系以及左本质谱σle(S)的连通分支与本质谱σe(T)的某些子集的相交关系,给出左本质谱σle(S)的连通分支与σle(T)相交的充分条件和充要条件.     

Operators Close to Unitary Ones and Their Function Models. I

We construct a function model for an operator in a Hilbert space close to an isometry. The model operator acts on the space of functions meromorphic inside and outside the unit disk. The functions from the space may be regarded as generalizations of Cauchy integrals of distributions, which gives a base for spectral analysis. We prove a theorem on the existence of such a model for one-dimensional perturbations of a unitary operator. Bibliography: 2 titles.     

Riemann-Stieltjes算子的本性范数

本文建立在从Hardy空间H~p到Zygmund型空间Z_μ的Riemann-Stieltjes算子I_(g,φ)和J_(g,φ)的有界性和紧性的特征的基础上,构造了H~p中一些检验函数,运用本性范数的定义与解析函数的性质,给出了算子I_(g,φ)和J_(g,φ)本性范数的估计.     

Essential Spectra of One-Dimensional Dirac Operators

The present paper estimates or explicitly calculates the essential spectra of one-dimensional Dirac operators L by introducing two one-parameter pencils of Sturm?CLiouville operators ?? ₁(??) and ?? ₂(??) for ${\lambda \in \mathbb{R}}$ , considering spectral problems of ?? _j (??) with a new spectral parameter, and examining the sets ${\{\lambda \in \mathbb{R} :\,\,\, 0 \notin \sigma_{\rm e}(\tau_j(\lambda))\}}$ for j = 1, 2. Applications are given to illustrate the scope of applicability.     

Fredholm Operators, Essential Spectra and Application to Transport Equations

In this paper the essential spectra of closed, densely defined linear operators is characterized on a Banach spaces under perturbations of n-strictly power compact operators. Further we apply the obtained results to investigate the essential spectra of one-dimensional transport equation with general boundary conditions and the essential spectra of singular neutron transport equations in bounded geometries.     

算子的Kato本质谱与Browder本质谱的连通分支

利用算子谱的精密结构的分析方法,给出算子S和T拟相似时Kato本质谱σK(S)的一个连通分支与σK(T)相交的充分条件和充要条件,同时证明了Browder本质谱σB(S)的每一个连通分支与Kato本质谱σK(T)的某些子集的交集是非空的.     

p-亚正常算子的本质谱与拟相似

本文证明了每一个ｐ－亚正常算子Ａ,都相应存在一个亚正常算子 ,使得Ａ与 有相同的闭值域点、相同的本质谱和谱．由此推出如果Ａ是ｐ－亚正常算子,Ｂ是任一有界线性算子,若存在有界线性算子Ｘ有稠值域,使得ＸＢ＝ＡＸ,则σ（Ａ）（Ｂ）此外还证明了,如果Ａ是ｐ－亚正常算子且 Ｒ（Ａ）闭或ＫｅｒＡ＝ＫｅｒＡ*, Ｂ是任一有界线性算子,Ａ与Ｂ拟相似,则ｅ（Ａ）（Ｂ）．