On a Subspace Perturbation Problem

Notes on the subspace perturbation problem for off-diagonal perturbations

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded off-diagonal perturbation is studied. To this end, the optimization approach for general perturbations in [J. Anal. Math., to appear; arXiv:1310.4360 (2013)] is adapted. It is shown that, in contrast to the case of general perturbations, the corresponding optimization problem can not be reduced to a finite-dimensional problem. A suitable choice of the involved parameters provides an upper bound for the solution of the optimization problem. In particular, this yields a rotation bound on the subspaces…

Unconditional bases of subspaces related to non-self-adjoint perturbations of self-adjoint operators

Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union $\bigcup_{j\in J}Δ_j$, $J\subseteq\mathbb{Z}$, of segments $Δ_j=[α_j, β_j]\subset\mathbb{R}$ such that $α_{j+1}>β_j$ and $$ \inf_{j} \left(α_{j+1}-β_j\right) = d > 0. $$ If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $\|B\|=:b<d/2$ then the spectrum of the perturbed operator $A=T+B$ lies in the union $\bigcup_{j\in J} U_{b}(Δ_j)$ of the mutually disjoint closed $b$-neighborhoods $U_{b}(Δ_j)$ of the segments $Δ_j$ in $\mathbb{C}$. Let $Q_j…

Semidefinite perturbations in the subspace perturbation problem

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded semidefinite perturbation is considered. A variant of the Davis-Kahan $ \sin2Θ$ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46] adapted to this situation is proved. Under a certain additional geometric assumption on the separation of the spectrum of the unperturbed operator, this leads to a sharp estimate on the norm of the difference of the spectral projections associated with isolated components of the spectrum of the perturbed and unperturbed operators, respectively. Without this additional g…

Sharpening the norm bound in the subspace perturbation theory

Let A be a self-adjoint operator on a Hilbert space H. Assume that σ is an isolated component of the spectrum of A, i.e. dist(σ,Σ)=d>0 where Σ=spec(A)\σ. Suppose that V is a bounded self-adjoint operator on H such that ||V||<d/2 and let L=A+V. Denote by P the spectral projection of A associated with the spectral set σ and let Q be the spectral projection of L corresponding to the closed ||V||-neighborhood of σ. We prove a bound of the form arcsin(||P-Q||)\leq M(||V||/d), M: [0,1/2)-->R^+, that is essentially stronger than the previously known estimates for ||P-Q||. In particular, the bound obt…

谱聚类的留一奇异子空间扰动分析

Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering

本文提出留一奇异子空间扰动理论，建立新的扰动上界，并据此对混合模型下谱聚类进行逐元素分析，在更弱信噪比条件下得到亚高斯混合模型的最优显式指数误差率。

On an estimate in the subspace perturbation problem

The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. In the approach presented here, a constrained optimization problem on a specific set of parameters is formulated, whose solution yields an estimate on the arcsine of the norm of the difference of the corresponding spectral projections. The problem is solved explicitl…

Preserving spectral properties of structured matrices under structured perturbations

This paper is devoted to the study of preservation of eigenvalues, Jordan structure and complementary invariant subspaces of structured matrices under structured perturbations. Perturbations and structure-preserving perturbations are determined such that a perturbed matrix reproduces a given subspace as an invariant subspace and preserves a pair of complementary invariant subspaces of the unperturbed matrix. These results are further utilized to obtain structure-preserving perturbations which modify certain eigenvalues of a given structured matrix and reproduce a set of desired eigenvalues whi…

矩阵、有界线性算子和解析算子值函数的谱扰动理论简述

A brief overview of spectral perturbation Theory

本文简要介绍了矩阵、有界线性算子和解析算子值函数的谱扰动理论，重点分析了特征值、特征向量和不变子空间的扰动界及相关理论。

Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator

Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L. All the bounds are given in terms of the norm of V and the distances between pairs of disjoint s…

The detectable subspace for the Friedrichs model

This paper discusses how much information on a Friedrichs model operator can be detected from `measurements on the boundary'. We use the framework of boundary triples to introduce the generalised Titchmarsh-Weyl $M$-function and the detectable subspaces which are associated with the part of the operator which is `accessible from boundary measurements'. The Friedrichs model, a finite rank perturbation of the operator of multiplication by the independent variable, is a toy model that is used frequently in the study of perturbation problems. We view the Friedrichs model as a key example for the d…

The singularly continuous spectrum and non-closed invariant subspaces

Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\mathfrak{H}_0$ is of codimension 1, we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ of $\mathbf{A}$ that are off-diagonal with respect to the decomposition $\mathfrak{H}= \mathfrak{H}_0\oplus\mathfrak{H}_1$. In particular, we prove the existence of a one-parameter family of dense non-closed invariant subspaces of the operator $\m…

Spectral dissection of finite rank perturbations of normal operators

Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/ characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of $T$. Under mild geometric conditions on the spectral measure of $N$ and some smoothness constraints on $K$ we show that the operator $T$ admits invariant subspaces, or even it is decomposable.