Perturbations of simple eigenvectors¶of linear operators (original) (raw)

On some perturbation inequalities for operators

Linear Algebra and its Applications, 1988

An estimate for the norm of the solution to the equation AX-XB = S obtained by R. Bhatia, C. Davis, and A. McIntosh for normal operators A and B is shown to be valid for a larger class. Some other inequalities in the same spirit are obtained, including a "sin0 theorem" for singular vectors. Some inequalities concerning the continuity of the map A + IAl obtained recently by Kittaneh and Kosaki are extended using these ideas.

Perturbation of operators and approximation of spectrum

Proceedings - Mathematical Sciences, 2014

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x) n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x) n. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x) n. The known results for a bounded selfadjoint operator, are translated into the case of a holomorphic family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of holomorphic perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here.

Spectral perturbation bounds for selfadjoint operators I

Operators and Matrices, 2008

We give general spectral and eigenvalue perturbation bounds for a selfadjoint operator perturbed in the sense of the pseudo-Friedrichs extension. We also give several generalisations of the aforementioned extension. The spectral bounds for finite eigenvalues are obtained by using analyticity and monotonicity properties (rather than variational principles) and they are general enough to include eigenvalues in gaps of the essential spectrum.

On matching distance between eigenvalues of unbounded operators

Constructive Mathematical Analysis, 2022

Let AA and ~AA~ be linear operators on a Banach space having compact resolvents, and let λk(A)λk(A) and λk(~A)(k=1,2,…)λk(A~)(k=1,2,…) be the eigenvalues taken with their algebraic multiplicities of AA and ~AA~, respectively. Under some conditions, we derive a bound for the quantity md(A,~A):=infπsupk=1,2,…∣∣λπ(k)(~A)−λk(A)∣∣,md⁡(A,A~):=infπsupk=1,2,…|λπ(k)(A~)−λk(A)|, where ππ is taken over all permutations of the set of all positive integers. That quantity is called the matching optimal distance between the eigenvalues of AA and ~AA~. Applications of the obtained bound to matrix differential operators are also discussed.

Error Estimates for the Eigenvalue Problem for Compact Operators

2013

The paper deals with approximations of the eigenvalues of compact operators in a Hilbert space by the eigenvalues of finite matrices. Namely, let (a jk) ∞ j,k=1 be the matrix representation of a compact A in an orthonormal basis, and A n = (a jk) n j,k=1. A priori estimates are established for the quantity sup µ∈σ(A) min λ∈σ(An) |λ − µ|, where σ(A) is the spectrum of A.