A spectrum theorem for perturbed bounded linear operators

doi:10.1016/j.amc.2008.01.008

Applied Mathematics and Computation

Volume 201, Issues 1–2, 15 July 2008, Pages 723-728

https://doi.org/10.1016/j.amc.2008.01.008 Get rights and content

Abstract

Using a generalized Sherman–Morrison–Woodbury theorem for linear operators, we establish a relation for the spectra of a bounded linear operator on a Banach space and its perturbation of a special structure.

Introduction

Recently, motivated by the spectral analysis of the Google matrix and related topics [4], [5], [2], a spectral perturbation theorem has been proved in [1] for some rank-k updated matrices of special structure, with the help of the classic Sherman–Morrison–Woodbury formula for the inverse of rank-k perturbed matrices. Some more general results about the algebraic relations between the characteristic polynomials of a matrix and its specially structured perturbation have been obtained in [3], based on a determinant identity for perturbed matrices. Such eigenvalue perturbation results can be applied to the spectral analysis of the Google matrix for the computation of the PageRank in the Google web search engine (see [7] for more about the Google matrix and the PageRank).

In this paper, we extend some of the above mentioned results from the finite dimensional case of matrices to infinitely dimensional linear operators. More specifically, we shall investigate the spectral problem for some specially perturbed bounded linear operators on general Banach spaces, using a generalized Sherman–Morrison–Woodbury theorem for linear operators that we shall derive in the paper.

In the next section, we give some preliminary results for the later use, and we also extend the classic Sherman–Morrison–Woodbury theorem from matrices to general linear operators defined on general vector spaces. In Section 3 we prove our main theorem. Some applications to more concrete problems will be discussed in Section 4, and we conclude in Section 5.

Section snippets

Generalized Sherman–Morrison–Woodbury formula

In this paper by a linear operator T from a vector space X into a vector space Y of the same scalar field we mean that its domain $D (T)$ is a vector subspace of X and $T : D (T) \subset X \to Y$ is a linear mapping on $D (T)$ . The subspace $R (T)$ of Y defined by $R (T) = {Tx : x \in D (T)}$ is called the range of T. Let T be a linear operator from X into Y and S a linear operator from Y into Z. Then we can define a linear operator $ST : D (ST) \subset X \to Z$ by $STx = S (Tx)$ for all $x \in D (ST) \equiv {x \in D (T) : Tx \in D (S)}$ . In particular, if T is a linear operator

The spectrum theorem

Now we can apply the results of the previous section to the spectral perturbation analysis of linear operators on Banach spaces. Let X be a complex Banach space and let $T : D (T) \subset X \to X$ be a linear operator. A complex number $λ$ is in the resolvent set, $ρ (T)$ , of T if the linear operator $λ I - T : D (T) \to X$ is one-to-one and onto, and its inverse $(λ I - T)^{- 1} : X \to D (T)$ is a bounded linear operator. If $λ \in ρ (T)$ , then $R_{λ} (T) \equiv (λ I - T)^{- 1}$ is called the resolvent of T at λ. If λ not in $ρ (T)$ , then $λ$ is said to be in the spectrum $σ$

Some applications

We give several applications of our main result in some more special cases.

Since matrices define bounded linear operators on Euclidean spaces under any given norms, a direct consequence of our main theorem reproduces the eigenvalue theorem below that has been obtained in [1], [3].

Proposition 4.1

Let k and n be two positive integers, and let A, U, and V be $n \times n$ , $n \times k$ , and $k \times n$ complex matrices, respectively. If there is a $k \times k$ complex matrix $Λ$ such that $VA = Λ V or AU = U Λ,$ then $σ (A + UV) \cup σ (Λ) = σ (A) \cup σ (Λ + VU) .$

The next result is a

Conclusions

In this paper, we obtained a spectral result for the perturbation of a bounded linear operator of some special kind, based on the new established generalized Sherman–Morrison–Woodbury theorem for the invertibility of perturbed linear operators. Some more concrete results have also been obtained directly. It is our hope that such new propositions can find applications in the spectral analysis of bounded linear operators in mathematical and physical sciences. It would be interesting to extend our

Acknowledgement

Supported by the National Natural Science Foundation of China under the Grant 10425105 and the National Basic Research Program of China under the Grant 2005CB321704.

References (7)

J. Ding et al.
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Appl. Math. Comput.
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J. Ding et al.
Eigenvalues of rank-one updated matrices with some applications
Appl. Math. Lett.
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J. Ding et al.
Characteristic polynomials of some perturbed matrices
Appl. Math. Comput
(2008)

There are more references available in the full text version of this article.

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