scholarly journals Linear resolvent growth of rank one perturbation of a unitary operator does not imply its similarity to a normal operator

2002 ◽  
Vol 87 (1) ◽  
pp. 415-431 ◽  
Author(s):  
Nikolai Nikolski ◽  
Sergei Treil
Keyword(s):  
Unitary Operator ◽  
Normal Operator ◽  
Rank One Perturbation ◽  
Rank One ◽  
Resolvent Growth

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

scholarly journals Application of a Rank-One Perturbation to Pendulum Systems

2020 ◽  
Vol 12 (5) ◽  
pp. 47
Author(s):  
Traor´e. G. Y. Arouna ◽  
M. Dosso ◽  
J.-C. Koua Brou
Keyword(s):  
Perturbation Theory ◽  
Strong Stability ◽  
Double Pendulum ◽  
Simple Pendulum ◽  
Rank One Perturbation ◽  
Stability And Instability ◽  
Rank One

From a perturbation theory proposed by Mehl, et al., a study of the rank-one perturbation of the problems governed by pendulum systems is presented. Thus, a study of motion of the simple pendulum, double and triple pendulums with oscillating support, not coupled as coupled by a spring and double pendulum with fixed support is proposed. Finally (strong) stability and instability zones are calculated for each studied system.

SINGULARLY FINITE RANK NONSYMMETRIC PERTURBATIONS ${\mathcal H}_{-2}$-CLASS OF A SELF-ADJOINT OPERATOR

2021 ◽  
Vol 9 (1) ◽  
pp. 140-151
Author(s):  
O. Dyuzhenkova ◽  
M. Dudkin
Keyword(s):  
Finite Rank ◽  
Scalar Product ◽  
Separable Hilbert Space ◽  
Point Spectrum ◽  
Adjoint Operator ◽  
Formal Expression ◽  
Rank One Perturbation ◽  
Rank One ◽  
Negative Space ◽  
First Time

The singular nonsymmetric rank one perturbation of a self-adjoint operator from classes ${\mathcal H}_{-1}$ and ${\mathcal H}_{-2}$ was considered for the first time in works by Dudkin M.E. and Vdovenko T.I. \cite{k8,k9}. In the mentioned papers, some properties of the point spectrum are described, which occur during such perturbations. This paper proposes generalizations of the results presented in \cite{k8,k9} and \cite{k2} in the case of nonsymmetric class ${\mathcal H}_{-2}$ perturbations of finite rank. That is, the formal expression of the following is considered \begin{equation*} \tilde A=A+\sum \limits_{j=1}^{n}\alpha_j\langle\cdot,\omega_j\rangle\delta_j, \end{equation*} where $A$ is an unperturbed self-adjoint operator on a separable Hilbert space ${\mathcal H}$, $\alpha_j\in{\mathbb C}$, $\omega_j$, $\delta_j$, $j=1,2, ..., n<\infty$ are vectors from the negative space ${\mathcal H}_{-2}$ constructed by the operator $A$, $\langle\cdot,\cdot\rangle$ is the dual scalar product between positive and negative spaces.

scholarly journals Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1544
Author(s):  
Lvlin Luo
Keyword(s):  
Invariant Subspace ◽  
Functional Calculus ◽  
Unitary Operator ◽  
Normal Operator ◽  
Multiplication Operator ◽  
Bounded Linear ◽  
Normal Operators ◽  
Subspace Problem ◽  
Invariant Subspace Problem ◽  
Special Case

Let T:H→H be a bounded linear operator on a separable Hilbert space H. In this paper, we construct an isomorphism Fxx*:L2(σ(|T−a|),μ|T−a|,ξ)→L2(σ(|(T−a)*|),μ|(T−a)*|,Fxx*Hξ) such that (Fxx*)2=identity and Fxx*H is a unitary operator on H associated with Fxx*. With this construction, we obtain a noncommutative functional calculus for the operator T and Fxx*=identity is the special case for normal operators, such that S=R|(S−a)|,ξ(Mzϕ(z)+a)R|S−a|,ξ−1 is the noncommutative functional calculus of a normal operator S, where a∈ρ(T), R|T−a|,ξ:L2(σ(|T−a|),μ|T−a|,ξ)→H is an isomorphism and Mzϕ(z)+a is a multiplication operator on L2(σ(|S−a|),μ|S−a|,ξ). Moreover, by Fxx* we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class BLeb(H)⊂B(H) such that T is Li-Yorke chaotic if and only if T*−1 is for a Lebesgue operator T.

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