Perturbation analysis of singular subspaces and deflating subspaces

Summary. Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Received July 26, 1994     

Perturbation of invariant subspaces

We investigate by how much the invariant subspaces of a bounded linear operator on a Banach space change when the operator is slightly perturbed. If E and F are the spectral projector frames associated with A and A + H respectively, we answer the natural question about how far the two frames are in terms of the perturbation H and the separation of parts of the spectrum of the operator A. These results depend on how to measure the difference between the two frames and how to measure the separation between parts of the spectrum. These two measures are introduced and analysed.     

Perturbation of spectra of operator matrices and local spectral theory

For A∈L(X), B∈L(Y) and C∈L(Y,X) we denote by MC the operator defined on X⊕Y by . In this article, we study defect set DΣ=(Σ(A)∪Σ(B))?Σ(MC) for different spectra including the spectrum, the essential spectrum, Weyl spectrum and the approximate point spectrum. We then apply the obtained results to the stability of such spectra (DΣ=∅) and the classes of operators C for which stability holds of MC using local spectral theory.     

Perturbation analysis of generalized singular subspaces

This paper, as a continuation of the paper [20] in Numerische Mathematik, studies the subspaces associated with the generalized singular value decomposition. Second order perturbation expansions, Fréchet derivatives and condition numbers, and perturbation bounds for the subspaces are derived. Received January 26, 1996 / Revised version received May 14, 1997     

On iterative refinements for spectral sets and spectral subspaces

Stewart (1971) and Demmel (1987) have proposed iterative procedures for refining invariant subspaces of Hilbert space operators and matrices respectively. In this paper, modifications are proposed for these procedures which facilitates their application to bounded Banach space operators. Under regularity conditions (which could include densely defined closed operators) it is shown that the modifications perform as well as or better than the procedures of Stewart and Demmel.     

Perturbation bounds for means of eigenvalues and invariant subspaces

When a matrix is close to a matrix with a multiple eigenvalue, the arithmetic mean of a group of eigenvalues is a good approximation to this multiple eigenvalue. A theorem of Gershgorin type for means of eigenvalues is proved and applied as a perturbation theorem for a degenerate matrix.For a multiple eigenvalue we derive bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix. These bounds assure that, provided the dimensionalities are chosen appropriately, the angles of rotation of the subspaces are of the same order of magnitude as the perturbation of the matrix.A numerical example is given.     

Stability of semigroups of operators and spectral subspaces

No abstract. February 17, 1998     

Perturbation of spectra and Krein extensions

Rendiconti del Circolo Matematico di Palermo Series 2 -     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

Resolvent convergence and spectral approximations of sequences of self-adjoint subspaces

This paper studies resolvent convergence and spectral approximations of sequences of self-adjoint subspaces (relations) in complex Hilbert spaces. Concepts of strong resolvent convergence, norm resolvent convergence, spectral inclusion, and spectral exactness are introduced. Fundamental properties of resolvents of subspaces are studied. By applying these properties, several equivalent and sufficient conditions for convergence of sequences of self-adjoint subspaces in the strong and norm resolvent senses are given. It is shown that a sequence of self-adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence. A sufficient condition is given for spectral exactness of a sequence of self-adjoint subspaces in an open interval lacking essential spectral points. In addition, criteria are established for spectral inclusion and spectral exactness of a sequence of self-adjoint subspaces that are defined on proper closed subspaces.     

Perturbation of invariant subspaces of the equations of elasticity: Spectral theory

This paper is a discussion of the perturbation of operators in certain of their invariant subspaces. The program is carried out by the study of the simple nontrivial example, the equations of linear elasticity.     

Sub-stable and weak-stable manifolds associated with finitely non-resonant spectral subspaces

In this work we study maps of a Banach space near a fixed point. We show the existence and uniqueness of a class of local invariant sub-manifolds of the stable manifold which correspond to a spectral subspace satisfying a finite non-resonance condition of order and an overriding condition of order (condition (3) of Theorem 1). We study the dependence of these invariant manifolds on a parameter that lies in a Banach space. We also show that a local weak-stable manifold that satisfies these two conditions is unique in the class of maps. The uniqueness is due to the fact that our method does not require a cut-off function. An infinite dimensional Banach space does not always admit smooth cut-off functions. Received July 13, 1998; in final form August 12, 1999 / Published online February 5, 2001     

Essential spectra of spectral operators

The various essential spectra of a linear operator have been surveyed byB. Gramsch andD. Lay [4]. In this paper we characterize the essential spectra and the related quantities nullity, defect, ascent and descent of bounded spectral operators. It is shown that a number of these spectra coincide in the case of a spectral or a scalar type operator. Some results known for normal operators in Hilbert space are extended to spectral operators in Banach space.     

Spectral measures and duality of spectral subspaces of contractions with slowly growing resolvent

We consider the class II of contracting operators T with spectrum on the unit circle r, acting on a separable Hilbert space and subject to the following restriction on the growth of the resolvent R_T(): We study the spectral subspaces _T(B) for T, corresponding to arbitrary Borel subsets of the circle r; in parallel we study a Borel measure _T(B) on r, adequate for _T(B) in the following sense:Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 203–206, 1977.     

Invariant subspaces for Banach space operators with an annular spectral set

Consider an annulus for some , and let be a bounded invertible linear operator on a Banach space whose spectrum contains . Assume there exists a constant such that and for all polynomials . Then there exists a nontrivial common invariant subspace for and .

     

Boundedness of adjoint bases of approximate spectral subspaces and of associated block reduced Resolvents

Block reduced resolvents are often employed in iterative schemes for refining crude approximations of the arithmetic mean of a cluster of eigenvalues and of a basis of the corresponding spectral subspace. We prove that if the bases of approximate spectral subspaces are chosen in such a way that they are bounded and each element of the basis is bounded away from the span of the previously chosen elements, then the corresponding adjoint bases are also bounded. We give an integral representation of the associated block reduced resolvent and show that under such a choice of the bases, the approximate block reduced resolvents are bounded as well. This is crucial in obtaining error estimates for the iterates of several refinement schemes. In the framework of a canonical discretization procedure for finite rank operators, appropriate choices of ises are given for various finite rank approximation methods such as Projection, Sloan, Galerkin, Nyström, Fredholm, Degenerate kernel. If the bases are not chosen appropriately, the error estimates may no longer hold and the iteration scheme may not be numerically stable. Examples are given to illustrate these phenomena     

Essential spectra through local spectral theory

Based on a nice observation of Eschmeier, this is a study of the use of local spectral theory in investigations of the semi-Fredholm spectrum of a continuous linear operator. We also examine the retention of the semi-Fredholm spectrum under weak intertwining relations; it is shown, inter alias, that if two decomposable operators are intertwined asymptotically by a quasi-affinity then they have identical semi-Fredholm spectra. The results are applied to multipliers on commutative semisimple Banach algebras.