Balmohan V. Limaye

Spectral computations for bounded operators

SPECTRAL DECOMPOSITION Genera Notions Decompositions Spectral Sets of Finite Type Adjoint and Product Spaces SPECTRAL APPROXIMATION Convergence of operators Property U Property L Error Estimates IMPROVEMENT OF ACCURACY Iterative Refinement Acceleration FINITE RANK APPROXIMATIONS Approximations Based on Projection Approximations of Integral Operators A Posteriori Error Estimates MATRIX FORMULATIONS Finite Rank Operators Iterative Refinement Acceleration Numerical Examples MATRIX COMPUTATIONS QR Factorization Convergence of a Sequence of Subspaces QR Methods and Inverse Iteration Error Analysis REFERENCES INDEX Each chapter also includes exercises

Computation of Spectral Subspaces for Weakly Singular Integral Operators

Abstract This paper deals with finding bases for finite-dimensional spectral subspaces of a bounded operator on the linear space of all complex-valued continuous functions defined on a compact Hausdorff space. This goal is achieved by computing an exact basis for a spectral subspace of an approximate operator which is not of finite rank. The theoretical framework allows a wide class of approximations, and a special emphasis is given to Kantorovichs singularity subtraction discretization of weakly singular compact integral operators. An application to Hopfs operator in the context of the transfer equation in stellar atmospheres illustrates the numerical computation of a bidimensional spectral subspace corresponding to a cluster of eigenvalues.

Accelerated spectral approximation

A systematic development of higher order spectral analysis, introduced by Dellwo and Friedman, is undertaken in the framework of an appropriate product space. Accelerated analogues of Osborns results about spectral approximation are presented. Numerical examples are given by considering an integral operator.

Solution of a Schrödinger equation by iterative refinement

A simple eigenvalue and a corresponding wavefunction of a Schrodinger operator is initially approximated by the Galerkin method and by the iterated Galerkin method of Sloan. The initial approximation is iteratively refined by employing three schemes: the Rayleigh-Schrodinger scheme, the fixed point scheme and a modification of the fixed point scheme. Under suitable conditions, convergence of these schemes is established by considering error bounds. Numerical results indicate that the modified fixed point scheme along with Sloans method performs better than the others.

A fixed point technique to refine a simple approximate eigenvalue and a corresponding eigenvector

Let T be a bounded operator on a Banach space X. Let λ0 be a nonzero simple eigenvalue of a ‘nearby’ operator T0 and let ⊘0 be a corresponding eigenvector. Several modified versions of a fixed point scheme are given for iteratively refining the initial approximations λ0 and ⊘0 of an eigenvalue λ of T and a corresponding eigenvector ⊘ Convergence of these schemes is proved by considering error bounds for the iterates. These bounds hold if a compact operator T is approximated in the norm or in a Collectively compact manner by a sequence (T0) of bounded operators, and λ0 and ⊘0 are eigenelements of Tn0 for a fixed n0 of ‘moderate’ size. Numerical examples are no included to illustrate the performation of various iteration schemes.

On the steps of convergence of approximate eigenvectors in the Rayleigh-Schrödinger series

SummaryError bounds are given for the iterative computation of the eigenvectors in the Rayleigh-Schrödinger series. These bounds remove the discrepancy in the theoretical behaviour and numerical results, noted by Redont, under the assumption of collectively compact convergence. As a particular case, it follows that the eigenvector in the iterative Galerkin method proposed by Sloan improves upon the eigenvector in the Galerkin method. This is illustrated by numerical experiments.

Korovkin-type approximation on C∗-algebras

THEOREM: Let A and B be complex C*-algebras with identities 1, and I,, respectively. Let (4.) be a sequence of positive linear maps with #.( lA) Q 1, from A to B, and d a C*-homomorphism from A to B. Then C = {a E A: 4.(a) -+

On the accuracy of the Rayleigh-Schrödinger approximations☆

(a), )“(a* 0 a) --t #(a* 0 a)) is a norm-closed *subspace of A and is closed under the Jordan product a o b = (ab + ba)/2 in A. If all (, and ( are Schwarz maps, then C is, in fact, a C*-subalgebra of A. Here -+ denotes the operator norm convergence, or the weak operator convergence, or the strong operator convergence. A modification of this theorem for convergence in the trace norm is also considered. Various examples are given to illustrate the theorem.

Finite-rank methods and their stability for coupled systems of operator equations

Improved upper bounds are given for the kth terms of the Rayleigh-Schrodinger series for the eigenelements of a perturbed operator. These are used to obtain error bounds, which are better than the previously known estimates, for the iterative refinements of computed eigenvalues and eigenvectors, in the case of a collectively compact approximation. If the perturbation operator is of a special kind, stronger results are available; these results show a shift in the convergence pattern of approximate eigenvectors as compared to that of approximate eigenvalues.

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

Let